Photoemission and optical investigation of the electronic structure of molybdenum and ruthenium
by Kenneth Albert Kress
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Physics
Montana State University
© Copyright by Kenneth Albert Kress (1969)
Abstract:
The photoemission and optical properties of vapor deposited films of molybdenum (Mo) and ruthenium
(Ru) have been measured at room temperature. The photoemission properties measured are in the
spectral range of the threshold (4.3 eV and 5.4 eV for Mo and Ru respectively) to 11.8 eV.
The optical properties are measured from 0.5 to 11.8 eV. The data from both Mo and Ru are found to
be consistent with, and analyzed by, the nondirect transition model. Corrections for the escaping
scattered electrons are included in the optical density of states (ODS) analysis. Peaks in the ODS are
found at E - EF = -0.5 eV, -1.6 eV, and -3.9 eV for Mo; and -0.4 eV, -1.3 eV, and a third tentatively
placed at -3.6 eV for Ru.
The ODS for E > Ep is determined by direct numerical inversion of the nondirect model expression for
the dielectric constant. Several peaks appear in the ODS of Mo for E > EF while only one is observed
in the ODS of Ru. The ODS of Mo and Ru is compared with the band calculations of Matthesis for
tungsten (Matthesis, 1965) and rhenium (Matthesis, 1966) respectively. The relation of the measured
ODS's to the explanations based on the electronic density of states for the anomalous isotopic mass
dependence of the superconducting transition temperature is discussed. For Mo the volume loss
function has a peak at 10.8 eV while the absorption coefficient has a minimum at 11.3 eV, which
correlates with a minimum in the quantum yield at approximately 11.0 eV. The energy distribution of
the photo-emitted electrons show slight structural changes in the spectral range, 10 to 11 eV. In the
same spectral region a similar but weaker correlation between structure in the loss function, absorption
coefficient, yield, and the energy distributions of Ru is noted. V
PHOTOEMISSION AND OPTICAL INVESTIGATION OF THE ELECTRONIC
STRUCTURE OF MOLYBDENUM AND RUTHENIUM
by
KENNETH ALBERT KRESS
A thesis submitted to t h e •Graduate Faculty in partial
fulfillment of the requirements for the degree
of
DOCTOR OF PHILOSOPHY'
inPhysics
Approved:
MONTANA STATE UNIVERSITY
B o z e m a n ,'Montana
December 1969
iii
ACKNOWLEDGMENT
The author wishes
to extend sincere thanks
J. L a p e y r e , his thesis advisor,
to Dr. Gerald
for guidance and constructive
criticism made throughout the course of this work.
His aid
was especially helpful during the preparation of this m a n u s ­
cript.
Discussions wit h Dr. A. J. M. Johnson,
Dr. N. M o i s e , Dr.
K . N o r d t v e d t , and Mr.
proved- valuable.
The contributions of Mr.
Dr. M. R u g h e i m e r ,
G . Stensland also
C . Badgley and Mr.
F . Blankenburg in the mechanical design, electrical design,
and the construction of part of the apparatus used in this
investigation was certainly appreciated.
The financial
support of the National Aeronautics and Space Administration,
the Air Force Office of Scientific Research,
State University are gratefully acknowledged.
and Montana
Special thanks
should be extended to my wife for the drafting of the various,
figures
and the typing of the manuscript.
I
iv
TABLE OF CONTENTS
Chapter
I
II
III
Page
I N T R O D U C T I O N .....................'...............
I
PHOTOEMISSION AND OPTICAL PROPERTIES
OF M E T A L S ............
5.
A. Optical Constants •
B . Photoemission ...............................
1 . Pho to e x c i t a t i o n .........................
2. Transport and E s c a p e ....................
3. Photoemitted E l e c t r o n s ................ ■ .
4. Quantum Yield
.....................
5. Scattered Electrons in the Energy
Distribution Curves • ....................
C . Optical Density of States Analysis
. . . .
D . S u m m a r y ............. .. . . . ..............
7
.11
12
16
17
19
EXPERIMENTAL PROCEDURES AND EQUIPMENT
34
.
19
23
32
A.
Photoemission Measurements
. . . . . . . . .
34
1. Basic Experimental Apparatus
and P r o c e d u r e s ............................
34
2. Vacu u m Equipment and Procedures
. . . .
37
3. Sample Preparation
....................
39
4. Energy Distribution Curve Measurement
. ' 39
5.. Quantum Yield
•. . . . ..............
.
41
B . Optical M e a s u r e m e n t s ........................ . 4 4
1. R e f l e c t o m e t e r .........................
.
46
C . Data R e p r o d u c i b i l i t y ......................
49
"
IV
MOLYBDENUM EXPERIMENTAL R E S U L T S ......... '. .
A. Optical Measurements
.......................
1.. R e f l e c t a n c e .......................
2. Dielectric. C o n s t a n t ........... • . . . ' .
3. Loss Functions
................
4. Alpha, (hv)n(hv), (hv )2
(hv). . . .
5. Sum R u l e s ................................
B. Photoemission Measurements .................. ■
1. Y i e l d ..................................
2. Energy Distribution Curves
' ■
. 3. Optical Density of States for
M o Iybdenum
. . . . •. . . -. . ". . . . '.
C . Summary
.............
53
53
.53
60
61
63
65
65
66
68
71
80
V
Table of Contents Continued
Chapter
V
,
RUTHENIUM EXPERIMENTAL RESULTS
Page
................
A. Optical M e a s u r e m e n t s ................... 82
1. R e f l e c t a n c e .............................
2. Dielectric Constant
....................
3. Loss Functions
; ......................
4. A l p h a , (hv)n(hv), (hv )2C 2 Chv)
. .. .
5. Sum R u l e s .............................. ■ .
B . Photoemission Measurements
..............
1. Quantum Y i e l d ...........................
2. Energy Distribution Curves
.............
3. Optical Density of States for
R u t h e n i u m ................ ■...............
C . S u m m a r y ............................. .. . . .
VI
INTERPRETATION AND CONCLUSIONS
..............
82
82
85
86
86
86
89
90
92
95
102
104
A. Optical Density of States and Band
Structure of Molybdenum and Ruthenium
. .
104
1. M o l y b d e n u m .......................
104
2. R u t h e n i u m ................................... 114
3. Nondirect and Direct Transitions in
M o l y bdenum and Ruthenium ................
118
B . The Optical Density of States and
Isotropic Mass Effect in Molybdenum
and R u t h e n i u m ...........
120
APPENDIX A: Kramers-Kronig Analysis
.........
126
APPENDIX B : Study of Cesium Covered
R u t h e n i u m ............................ 132
LITERATURE CITED
....................... ..
. .
138
vi
LIST OF TABLES
Table
I
'II
III
Page
Comparison of Optical Constants of
Molybdenum of Present Study with those
Measured by Others
............................
59
Rigid Band Model Applied to the Theoretical
Calculation of the Density of States
of Cr and W
........................... .. . . .
108
Rigid Band Model Applied to the Experimental
Observation of the Density of States
of Cr and M o .....................................
HO
vii
LIST OF FIGURES
Figure
1
Page
Hypothetical D e n s i t y of States for a Transition
M e t a l . Shaded Area Represents Occupied or
Initial States and Open Area Represents
Unoccupied or Final States
.............
13
Free Electron Threshold Function in Units
of the Photoelectric Work F u n c t i o n , c f > .........
18
3
Electron-Electron Scattering Event ..............
21
4
Hypothetical EDC Structure Plot
27
5
Schematic of Basic Apparatus Used in
Photoemission E x p e r i m e n t ................ ..
2 ■
6
7
8
9
10
11
12
13 '
. . .............
. . .
35
Schematic of Vacuum Chamber for Photoemission
S t u d i e s ............................................
38
Quantum Yield of Gold Compared to that
taken by Krolikowski
...........................
45
Schematic of Reflectometer used for
Reflectance Measurements
.......................
47
Direct Tracing of Experimental Energy
Distribution Curves for hv = 10.2 eV from
Experimental C h a r t .................
50
Summary of Reflectance Measurements
for M o l y bdenum ............................ • . . . . . '
54
Reflectance of M o l y bdenum where i is the
Angle of .Incidence
..............................
58
Dielectric Function of Molybdenum
58
. . . . . . . .
Loss Functions for Mo l y b d e n u m ■ . . . .
.
. ...
. . . .
.
62
14
Absorption Coefficient a for Molybdenum
62
15
Optical Functions (hv)n(hv) and (hv)^ £2 (hv)
for M o l y b d e n u m ......................................... 64
©
viii
List .of Figures Continued
Figure
16
Page
Interband Njg, Surface Plasma N g p , and Volume
Plasma Nyp Sum Rules for M o l y b d e n u m ...........
64
17
Quantum Yield for M o l y b d e n u m ........... ..
.
67
18
Normalized Energy Distribution Curves for
M o l y bdenum Plotted Versus E - < j ) .................. •
69
Arbitrarily Normalized Energy Distribution
Curves of Molybdenum Plotted Versus E-hv
...
70
20
Structure Plot for M o l y b d e n u m ..................
72
21
Normalized Energy Distribution Curves
Multiplied by (hv)n(hv) of Molybdenum
Versus E - h v .............................. ..
73
19
22
N ^ ^ (E) for Various Photon E n e r g i e s ........... ' 7 4
23
Zeroth Approximation to the Optical Density of
States Used to Estimate the Scattered Electron
Contribution to the Energy Distribution Curve
of M o l y b d e n u m ............................■ . . . . '
76
Measured, Corrected and Scattered Energy
Distribution Curves of Photoemitted Electrons
from Molybdenum at hv = 8 .0 eV
. .'...........
78
Measured, Corrected and Scattered Energy
Distribution Curves of Photoemitted Electrons
from M o l y bdenum at hv = 11.0 eV
79
Optical Density of States of Molybdenum
where the Dashed Line is the Average Value
for the Density of States above the
Fermi E n e r g y .......................................
81
Reflectance of Ruthenium where i is the
Angle of Incidence
.......................
84
24
25
26
27
. . .
28
Dielectric Constant of M o l y b d e n u m ........... ..
.84
29
Loss Function of Ruthenium
87
.....................
ix
List of Figures Continued
Figure
30
.31
Page
Absorption Coefficient a for Ruthenium
.........
87
2
Optical Functions (hv)n(hv) and (hv)
for R u t h e n i u m .....................................
88
Interband Njg, Surface Plasma N g p , and Volume
Plasma Nyp Sum Rules for Ruthenium
. . . . . . .
88
33
Quantum Yield of Ruthenium
91
34
Normalized Energy Distribution Curves of
Ruthenium versus E-hv
. .
93
35
Structure Plot for Ruthenium
94
36
Effective Density of Final States
for R u t h e n i u m .......................................
32
37
38
39
- 40
41
......... ' .........
..................
97
Zeroth Approximation for the Optical Density
of States used to Estimate the Scattered
Electron Contribution to the Energy
Distribution Curves of Ruthenium
...............
Measured, Corrected and Scattered Energy
Distribution of Photoemitted Electrons, from
Ruthenium at hv = 8.0 e V .................... ..
.98
. - IOQ
Measured, Corrected and Scattered Energy
Distribution of Photoemitted Electrons from
Ruthenium at hv = 11.0 e V ....................... 101
Optical Density of States for Ruthenium
. . . .
The Optical Density of States of Molybdenum
is Shown by the Dashed Line.. ■ The Dashed
Line Indicates the Density of States
Estimated from Matthesis Tungsten (Wj) Band
Structure Calculations ..............■.
. .
103
106
X
List of Figures Continued
Figure
42
43
.44
45
46
47
"
The Optical Density of States from Eastman
(1968) (Solid Line) and the Density of
States Estimated from Connolly (1968) Band
Structure Calculation for Chromium
(Dashed Line)
.................. ■.......... ..
Page
. .
109
Comparison of the Optical Density of States
„ of M o l y bdenum (present study) and Chromium
. (Eastman, 1968)
....................... ........... Ill
The Optical Density of States of Molybdenum
Compared with the Density of States Estimated
- from M a t t h e s i s ' Tungsten (Wj j ) Band Structure
Calculation.
Note' the Energy Scale and the
Position of the Fermi Level of M a t t h e s i s '
W n Calculation was Arbitrarily Adjusted
. . .
113
The Optical Density of States (Solid Line)
and the Density of States of Ruthenium
Estimated from M a t t h e s i s 1 Rhenium Band
1
Structure Calculation (Dashed Line)
...........
115
The Optical Density of States of Ruthenium
Compared with the Density of States
Estimated from M a t t h e s i s ' Calculation of
the Density of States of Nonmagnetic Iron
from Wood's Energy Band Calculations
. .
.
117
Optical Transition Strength Function Calculated
from Infrared Optical Constants of Molybdenum
Measured by Kirillova et a l .................... 123
A-I
Logic Diagram of Computer Program used
to Calculate Optical Constants
................
, ...
130
B-I
Quantum Yield of Ruthenium with Approximately
Two Monolayers of C e s i u m ......................... 134
B-2
E D C s of Photoemitted Electrons from
Cr covered Ru
.......................
135
ABSTRACT
The photoemission and optical properties qf vapor
deposited films of molybdenum (Mo) and ruthenium (Ru) have
been measured at room temperature.
The photoemission
properties measured.are in the spectral range of the threshold
(4.3 eV and 5.4 eV for Mo and Ru respectively) to 11.8 eV.
The optical properties are measured from 0.5 to 11.8 eV.
The
data from both Mo and Ru are found to be consistent with, and
analyzed by, the nondirect transition model.
Corrections for
the escaping scattered electrons are included in the optical
density o f states .(ODS) analysis.
Peaks in the ODS are found
at E - Ep = -0.5 eV, -1.6 eV, and -3.9 eV for Mo; and . -0.4 e V ,
-1.3 eV, and a third tentatively placed at -3.6 eV for Ru.
The ODS for E > Ep is determined by direct numerical inversion
of the nondirect model expression for the dielectric constant.
Several peaks appear in the ODS of..Mo for E > Ep while only
one is observed in the ODS of Ru.
The ODS of Mo and Ru is
compared with the band calculations of Matthesis for tungsten
(Matthesis, 1965) and rhenium (Matthesis, 1966) respectively.
The relation of the measured O D S 1s to the explanations based
on the electronic density of states for the anomalous isotopic
mass dependence of the superconducting transition temperature
is discussed.
For Mo the volume loss function has a peak at ■
10.8 eV while the absorption coefficient has a minimum at 11.3
eV, which correlates with a minimum in the quantum yield at
approximately 11.0 eV.
The energy distribution of the photoemitted electrons show slight structural changes in the
spectral range, 10 to 11 eV.
In the same spectral region a
similar but' weaker correlation between structure in the loss
function, absorption coefficient, yield, and the. energy
distributions of Ru is n o t e d .
I.
INTRODUCTION
The electronic structure or character of the outer shell
electrons determines most of the observed properties of
condensed matter.
electrons
The theory of s- and d - like outer shell
in the transition metals has been investigated by
detailed band structure calculations within the one-electron
or independent particle a p p r o x i m a t i o n .
These calculations'
are now possible largely because of the availability of h i g h ­
speed computers.
Much experimental work on the electrons
in
metals has been done by methods such as de Haas-van A l p h e n ,
cyclotron resonance, magnetoacoustic, high field m a g n e t o ­
resistance,
and anomalous skin effect measurements.
These
measurements only probe the state in the immediate vicinity
of the Fermi energy.
Electronic states below the Fermi energy
may be investigated by soft x-ray absorption a n d . e m i s s i o n ,
and Auger electron emission p r o duced by ion bombardment.
The
latter two methods are capable of giving information c o n ­
cerning the gross features of the occupied outer shell or
valence electronic structure, but lack sufficient resolution
to yield detailed information that can be compared with the
theoretical results.
The band structure calculations for the transition and
noble metals
metals
indicate the electronic structure of these
is on the order of 10 eV wide.
This energy range is
-2investigated conveniently by photoemission and optical
studies using vacuum ultraviolet r a d i a t i o n .
Detailed e x p e r i ­
mental information concerning the outer shell electrons for
the noble metals
over a wide range of energy was first
demonstrated by the optical investigations
Philipp
(1962).
of Ehrenreich and
Their results were correlated with detailed
band calculations by assuming conservation of the Bloch sta't’e
wave vector, 5, in the optical excitation process.
afterwards,
Berglund and Spicer
(1964)
Shortly
demonstrated that more
detailed correlations could be obtained between band structure
predictions and experimental photoemission observations in the
metals copper
(Cu)
and silver
(Ag), provided conservation of
the Bloch wave n u m b e r , or crystal momentum selection rule, was
ignored for emission from d - like states.
The model of Spicer
and Berglund was called the nondirect model in contrast with
the standard Bloch wave-number-conserving direct transition
model.
The subsequent investigation of Spicer and his co- ,
workers soon produced results which indicated nondirect
transitions were dominant in the photoemission data of the 3d
transition metals nickel
1967),
and cobalt
(Co)
(Ni),
iron
(Fe)
(Yu and Spicer,
were extended to the 3d metal chromium
Kress
(1968) .
(Blodgett -and Spicer,
1967).
These results
(Cr) by Lapeyre and
The 3d metal Mn was completed and the whole, 3d'
series reinvestigated by Eastman
(1969).
Before the nondirect
-3optical transition model could be associated with the d-band
structure of metals
in general,
it became apparent that the
systematics of the electronic structure in the 4d and/or Sd
transition metals should be investigated.
Photoemission and optical investigations of the density
of states of the 4d and Sd transition metals could also give
information on the validity of the rigid band m o d e l .
In its
simplest form the rigid band model for a given transition"
metal period
(e.g. the 3d period vanadium
(V), Cr, manganese
(Mn), etc.) predicts only the position of the Fermi energy
and not the structure changes in the density of states as one
proceeds across the periodic table.
group
(e.g.
Furthermore,
group VI B : Cr, molybdenum
for a given
(Mo), and tungsten
(W))
neither the position of the Fermi energy nor the structure of
the density of states should change according to the rigid
band model.
The details of these predictions are expected to
fail when the crystal structure of the adjacent metals changes
The transition metals have an anomalous isotopic mass
dependence in their superconducting transition temperatures
(Garland,
1963).
ruthenium
(Ru)
In particular the 4d transition metals
and Mo deviate from the normal isotopic mass
effect observed in simple metals.
The 4d transition metals
Ru and Mo were chosen for .this investigation to add further
knowledge to the systematics of the electronic structure of
-4 the transition metals beyond the 3d series and to investigate
possible anomalous structure in the d-bands which could
account for their superconducting transition temp e r a t u r e s .
II.
PHOTOEMISSION AND OPTICAL PROPERTIES OF METALS
Measurements
of the kinetic energy distribution of
photoemitted electrons as a function of the incident photon
energy constitute the greatest single source of information
obtained from photoemission experiments.
bution curves
(EDO's)
The energy d i s t r i ­
of the photoemitted electrons from a
metal may be related to the electronic structure of the
photoemitting material according to the following simplified
model.
The photoemission process may be separated concep­
tually into three stages:
photoexcitation of the electron by
atomic absorption of the incident photon,
excited electron through the solid,
surface boundary of the metal.
transport of the
and escape across the
The incident photons trans­
mitted through the surface of the metal are typically
absorbed within a few hundred Angstroms of the surface.
The
excited electrons produced by the photon absorptions can
escape the metal provided they have the proper momentum and
have kinetic energies
greater than the vacuum level,
i.e.
the Fermi energy plus the photoelectric work function.
transport of the excited electrons
elastic
(e.g. electron-phonon)
electron)
During
towards the surface, nearly
and inelastic
collisions may.take place.
(e.g.
electron-
Estimates of the mean
inelastic scattering lengths for the photoexcited electrons
in
"6 “
transition metals are a few tens of Angstroms
1968).
(Eastman,
F i n a l l y , the electrons will escape the metal provided
their component of momentum perpendicular to the surface is
sufficient to overcome the photoelectric work f u n c t i o n .
If transport and surface escape effects of the photoemitted electrons
can be neglected or properly handled,
kinetic energy of the photoelectrons
the
is simply related to its
initial and final energy by the conservation of energy
principle.
From the above model,
the EDO's of photoemitted
electrons as a function of photon energy are related to the
relative transition probability between the initial and
final states.
The experimental transition probability can ■
then be directly related to the electronic structure or
density of states at initial and final e n e r g y .
Measurements
of the reflectance as a function of the
incident photon energy may also be used to obtain information
about the electronic structure.
The optical "constants"
derived from the measured reflectance as a function of the
photon e n e r g y , E = hv can be related to the total optical
transition probability of all states separated by the energy
hv.
The energy of the photoexcited electron produced by
photon absorption is not determined by optical studies,
therefore the absolute energy of the initial or final states
participating in the transitions cannot be determined.
Since
-7the optical constants are directly related to the sum of all
optical transitions which conserve e n e r g y , they provide less
precise information about electronic structure of metals than
photoemission functions.
When both photoemission a n d 'optical studies are performed
on the same metal,
there is sufficient complementarity and
overlap of the data to allow checks and extensions of the
information o b t a i n e d .
Photoemission and optical studies are
combined in the present study to produce a more comprehensive
study of the electronic structure of metals.
A.
Optical Constants
\
Beginning with Maxwell's equations and the constitutive
equations,
the interaction of radiation with matter can be
described by a set of optical "constants" which vary with the
photon energy.
constants
There are three sets of complex optical
generally defined in optical measurements.
of these sets is equivalent to any other set.
Any one
These constants
are the complex refractive index N (hv), complex dielectric
constant e (hv) and complex conductivity a (hv) where,
in cgs
units:
N (hy) = n (hv) + ik(hv),
( 2 - 1)
e(hv) = E1Chv) + i E2 (hv).
(2-2)
-8and
cr(hv) = CT1 O v )
+ i a 2 (hv) .
(2- 3)
With the aid of Maxwell's equations and the constitutive
equations,
these constants may be related to each other
(Pines, 1964)
N2 (hv) = E(Iiv) = I + 21 ^ h-vJ
.
(2-4)
The real and imaginary parts of this equation can be set
equal and relationships between these portions' derived,
E 1 (hv) = n 2 (hv)
- k 2 (hv) = 1 - 2
o^/v
(2-5)
.
(2 - 6)
and
e 2 (hv) = 2n (hv) k (hv) = 2 o-^/v
Another optical constant that is considered is the
absorption coefficient a (hv).
The absorption coefficient is
.defined as the light intensity change per unit distance per
unit intensity and is simply related to k (hv) by
a(hv) - 4 r t CjiyJ-
(2-7)
where X is the wavelength of the incident radiation.
The imaginary part of the dielectric constant,
E 2 (hv),
.is directly related to the total optical absorption probability.
The relation between & 2 (hv) and the total number of optical
excitations has been derived by Berglund and Spicer
(1964).
-9 Assuming monochromatic incident radiation and the correspon­
dence principle their derivations
shows that the total number
of transitions per second is proportional to
the photon energy squared,
multiplied by
i.e.
hv
(hv)
62 (hv) oc
( 2- 8 )
P if dE
.0
where
is the rate of transitions of electrons
initially
with energy E-hv to energy E and the zero of the energy scale
is at the highest energy of the occupied levels.
e x p l i c i t 'expression for P ^
transition model
Using the
developed later for the nondirect
(cf E q . (2-19)),
the above equation may be
written as
hv
'(hv) 2 ^ 2 (bv) = A
where A is .a constant and
and
are the optical
density of initial and final states respectively.
function
(hv)
(2-9)
I?pt (E-hv) N^pt (E) dE
The optical
82 (hv) will be called the optical transition
strength function and is important in the determination of
N°p t (E).
The optical dielectric constant
e(hv) can be used to
determine the photon energy at which the properties of the
metal deviate from free electron-like behavior.
Ehrenreich and Philipp
(1962) have shown that
-For example,
£
-10£ (hv) =
(hv) + E ^ (hv)
( 2- 10)
,
where E^ (hv) represents the interband transitions and E ^ (hv)
represents
intraband transitions.
The intraband portion has
the same form as results .from the classical Lorentz model
for electrons
in a metal,
i.e.
Ef (hv) = I - v^/ [mv(v + i)]
where
2
( 2- 11 )
,
2
v = 2Ne /m, N = e l e c t r o n s / v o l u m e , m is the mass of the
P
-
electron,
and m/r = effective damping constant.
The real part
of the free electron-like portion of the dielectric constant,
E^(hv), monotonically approaches unity while the imaginary
part falls monotonically to zero as v increases.
The energy of collective excitations may be determined
from the optical constants.
It has been shown that the
general condition for a plasma oscillation at frequency ^
is E (ft) = 0 (Pines,
where
1964).
This condition yields
+ if
P
is the plasma frequency and T describes the damping.
When there is little damping,
plasma resonance becomes
or ^ ^ p ^
E^ (oj^) = I.
~
= w
the condition for
Then Im 1/e, the volume
loss function approaches a peak value of l/e 7 (a) ) at w .
general,
In
a plasma resonance can be distinguished from inter-
band effects which can also produce structure in Im .1/e by
noting that in the former case, both e^ and e^ are small near
-11the maximum in the loss function and have a positive and
negative slope respectively
(Philipp, 1965).
ment may be made concerning peaks
A similar s t a t e ­
in the surface plasma loss
.function Im I/(e + I) correlating with the surface plasma
excitations
(Ritchie, 1967).
Several sum rules are associated with the optical
constants.
Three such sum rules which define the effective
number of electrons per atom participating in interband
transition, volume plasmon
(Nozieres and Pines,
surface plasmon excitations
(Ritchie,
1959),
and
1968) below the energy
hv are given respectively by
N j b (hv)
____ m ___
Zir2N e V
hv
e 2 (E) EdE
,
( 2 - 12)
o
hv
NypChv)
Zir2N e V
[-Im(l/e(E))] EdE
(2-13)
)] EdE
[ - Im (
^e(E)+1
(2-14)
o
and
hv
(hv)
Tr2N e V
'o
where m is the free electron mass and N is the number of atoms
per unit volume.
B.
Photoemission
The results of the present photoemission study are found
to be consistent with and will be analyzed by the nondirect
optical transition model
(Berglund arid S p i c e r , 1964).
Other
-12models for optical transition in solids have been proposed
(a review of two other models is given by Berglund,
1964) but
have predicted different effects than those observed in the
present investigation.
Only key results relevant to the
nondirect model and the interpretation of the present study '.
will be presented h e r e .
The reader is referred to the cited
references for expanded discussions
and more detailed d e r i v a ­
tions.
I.
Photoexcitation.
The photoemission properties of a
metal may be interpreted in terms of its electronic structure
by considering specific, models for the three stages of p h o t o ­
emission of electrons
outlined a b o v e .
the initial photoexcitation process
In the nondirect model,
is treated by an applica­
tion of first order time dependent perturbation theory.
the electrons
length
If
in a metal are perturbed by a w e a k , long w a v e ­
(compared to atomic d i m e n s i o n s ) , electromagnetic f i e l d ,
the perturbing Hamiltonian is given by
H
= e/2mc
I P i e^ C r i)
where p^ is the momentum operator of the i ^
'
(2-15)
electron and ■
it(r^) is .the vector potential of the perturbing radiation
field.
In metals, the occupied initial states and unoccupied
final states of the electron
(sometimes called valence and
conduction states respectively)
are. quasi-continuous with
-13energy as displayed in Fig.
I.
H e n c e , summing over the
N .(E-hv)
Fig.
I.
Hypothetical Density of States for
a Transition M e t a l . Shaded Area
Represents Occupied or Initial
States and Open Area Represents
Unoccupied or Final States.
initial s t a t e s , assumed distributed with a density
energy,
E ^ = E
per unit
and explicitly putting in energy conservation,
and E^ = E - hv
i.e.
(the zero of the energy scales is
assumed to be the Fermi energy as shown in F i g . I), the
standard perturbation theory result for the rate of transi­
tions from energy E - hv to E
(Messiah, 1962)
is
-14-
if
Ztt
CY
H
Y .>
i
N i (E-Hv)Nf (E)
(2-16)
M zN f (E-Hv)Nf (E)
where N f (E-Hv)
(2-17)
is the density of initial states at energy
E - H v , N f (E) is the density of final states at energy E , Y f
and Yf are the exact initial and final states wave functions
r e s p e c t i v e l y , and the matrix element is abbreviated as M
From E q . (2-17)
.
it is seen that to first order the number of
photoexcited electrons to energy E depends upon three factors:
a matrix element, M
2
, the density of final states, N f (E), and
the density of initial states, N f (E-Hv).
It is difficult to obtain a theoretical expression for
the matrix element, M
2
, since the exact ground state and
excited state wave functions are u n k n o w n .
The matrix elements
contain all the selection rules determined by the symmetry
and spin states of the exact initial and final state wave
functions coupled by the interaction Hamiltonian and was
expected to be rapidly varying w ith respect to energy.
Contrary to this expectation,
(Berglund and Spicer,
1964)
the initial experimental results
indicated that M
as if it was nearly constant.
2
could be treated
It was later realized that if
the matrix element was proportional to a product of functions
which depend on the initial and final state energy of the
-15electrons,
E - hv and E r e s p e c t i v e l y , variations of the matrix
elements could not be distinguished .from the effects due to
unperturbed ground state density of states
(Spicer, 1967).
This assertion becomes clear if the postulated matrix element
dependence is substituted into E q . (2-17) .
P i f (E,E-hv)
=
(E-hv)N f (E-hv)M f (E)Nf (E)
= B N iP t (Erhv)NfP t (E)
(Z-IS)
(2-19)
is proportional to the photon flux and the definition
I
N fP t (E-hv)
= M f (E-Iiv)Nf (E-hv)
(2 - 20)
and
(2-21)
N fP t (E) = M f (E)Nf (E)
have been u s e d .
The experimentally observable densities of states N ? ^
(E-hv)
and
(E) are called the optical density of initial
and final states respectively.
the optical density of states
state density of states
Spicer
(1967) has postulated
(ODS) may differ from the ground
(DS) due to relaxation effects that
occur during optical excitation.
Others have estimated matrix
element variations due to the character of the d.-band wave
function
(Cuthill et a l ., 1967).
Optical and/or photoemission
investigations study the optical transition probability and do
not study the matrix elements directly.
Effects due to the
-16ground state density .of states and those due to matrix element
variations cannot be unambiguously distinguished by optical
excitation.
2.
Transport and E s c a p e .
excited electrons
The transport of the photo-
in a metal may be approximately treated by
assuming that the number of electrons excited at a depth x
below the surface of the metal is proportional to., the, absorbed
light intensity a( h v ) e a Chv )x at x ,
The subsequent probability
of the photoexcited electron reaching the surface without
scattering is further assumed to be proportional to e x Z^(E)
where L(E)
is the energy dependent, mean scattering length of
the excited electrons with energy E
(Berglund and Spicer,
1964).
Averaging these effects over all depths x, the total p r o b a ­
bility of a photoexcited electron with energy E migrating to
the surface of the metal without being scattered is p r o p o r ­
tional to
(2- 22)
o
■ a (hv) L(E)
1+a (hv) L(E)
L(E)
•
(2-23)
is the effective escape length and will become degenerate
with the electron-electron scattering length in the approximate
treatment assumed in the following development.
If the above derivation is done for a three dimensional
solid,
a function.K(a,
L , T) appears as a factor in the
-17transport term (Berglund and S p i c e r , 1964).
K - is a very
slowly varying function and is bound between 0.5 and 1.0 for
all a, L, and T.
This factor will be neglected in the
present study.
The proba b i l i t y of the excited electrons
escaping across
the surface boundary of the metal into the vacuum is treat,ed
in the calculation of the threshold function, T .
In general
T is a complex function of the bul k and surface properties of
the metal and is dependent on the details of the scattering
mechanisms
(Stuart and Wooten,- 1967) .
Insight into the
effects of the threshold function can be gained be considering
the threshold function derived for classical noninteracting
electrons with energy E and isotropic velocity distribution
in a step potential well of depth
(Berglund and Spicer,
1964),
0
Tf(B)
= .{
1
' E < <f)
_ _ _
’%(1 - /57BT
'B =
(2-24)
(2-25)
.
It is apparent from Fig.
2 that electrons with.energy less
■
than <j), the photoelectric work function, will not escape the'"'metal.
T^(E)
is a monotonically increasing function for
>
E > cf> and is most rapidly varying for E ^ c f > ..3..
Photoemitted E l e c t r o n s .
■
An expression for the number
of electrons photpemitted from the metal is obtained by
multiplying the probability of transport and escape of -the .
V4 ,
■18-
0.50
0.25
0.00
Fig.
2.
Free Electron Threshold Function in
Units of the Photoelectric Work
F u n c t i o n , cj>.
electrons by their probability of excitation,
N(E-Cf),hv)
cc
or
Ta(hv)L(E)N°pt (E-hv)N°p t (E)
------------ ±---------- £------ ,
I + a (hv) L (E)
where the energy argument of the photoemitted electrons
E - <j) since the electrons
(2-26)
is now
lose the energy <j> escaping from the
metal.
It is useful to normalize the above expression for photoemitted E D C 1s to the total number of photoexcited electrons/
absorbed photons.
absorbed photons
strength function
The total number of photoexcited electrons/
is proportional
(cf E q . (2-8))
to the optical transition
and the normalized
-19A T a (hv)L(E)N° p t (E-hv)N^p t (E)
N(E-((),hv) = ----------------------- ----------
[I + a(hv)L(E)] (hv)
where A is a constant.
,
'
(2-2 7)
Eg(Hv)
Since the optical transition strength
is an experimentally measured quantity,
the effects of the
exact matrix element are included in the normalization.
4.
Quantum Y i e l d .
The absolute quantum yield of the
metal is defined as the number of photoemitted. electrons per
absorbed photon.
Using the expression for the, normalized E D C ,
the yield is the sum of all the photoemitted electrons
normalized to the total number of photoexcited electrons or
hv
N (E - cj>,hv) dE
Y (hv) =
( 2 - 28)
; 4)
5.
Scattered Electrons
in the Energy Distribution
C u r v e s . .Multiple nearly elastic electron-phonon and inelastic
electron-electron scattering events of photoexcited electrons
have been considered by detailed Monte Carol calculations
(Stuart and Wooten,
1967).
These calculations were done
assuming an isotropic velocity distribution of the p h o t o ­
excited and scattered electrons
in addition to .-energy
dependent mean scattering lengths.
The multiple scattering
calculations indicate that the effects of the elastic
scattering events may be summarized in the threshold function
so that T - T (E).
-20The multiple scattering calculations also indicate that
inelastic scattering events which result in energy losses of
several eV per collision produce the dominant observable
effects on the E D C 's .
Since the photoelectric work functions
are typically several eV and the photon energy is limited by
the spectral transmission of the window on the experimental
chamber,
only a small fraction of the electrons which are
scattered possess sufficient energy to escape the m e t a l .
The
scattered electrons which do manage to escape the metal will
be called s e c o n d a r i e s .
An approximate analytical expression for photoemitted
once scattered secondary electrons has been developed
(Berglund and S p i c e r , 1964).
This analytical expression was
found to be an adequate description of scattering processes
in a metal by the more eloquent Monte Carlo multiple
scattering calculations
of Stuart and Wooten
but the lowest energy photoemitted electrons.
(1967)
for a l l c
Since the
analytical expression for secondaries has been shown to be
a reasonable approximation,
it will be used for the analysis
of the measurements performed in the present study.
The .analytical expression of Berglund and Spicer for
escaping scattered photoexcited electrons
is derived by
assuming only inelastic electron-electron scattering events
are significant. ■ The desired expression is obtained by
-21considering the scattering e v e n t .pictured in F i g . 3.
Fig.
3.
Electron-Electron Scattering
Event.
If constant matrix elements are assumed,
the probability of
the electron-electron event may be calculated by finding the
overlapping phase space for all
standard manner.
, 6 , and 6
in the
Assuming f u r t h e r , the scattered electrons
have an isotropic velocity distribution, a(hv)L (E) << I and
L (E) /L (E1) << I if -E1 > E , the expression for the energy
distribution of the photoemitted electrons
is modified by the
inclusion of a small term containing the electron-electron
scattering p r o b a b i l i t y , i.e. E q . (2-2 7) becomes
-22I0P t
A T (E) a (hv) L (E) N?pt (E-Iiv)N^t'(E)
N (E- <j>,hv)
[I + a(hv)L(E)](hv )2
£2 (hv)
opt
hv P(E,E')N?p t (E'-hv)N^p L (E')dE'
-
P(E')
'
I •+
T°p t
(2-29)
(E-Ev)N^pt (E)
= So (E,hv)N°p t (E-hv)N°p t (E)[! + S i (EjEv)]
(2-30)
where
'0
(E1)N^pt (E1+ E t-E)dE 1
P ( E jE t) = N^ pt (E)'
(2-31)
- ( E t-E)
E'
(2-32)
P (E,E ') dE
P ( E t) =
_0
So (EjEv) =
A T(E)a(hv)L(E)
I + a (Ev)L (E)
(2-33)
Ev P ( E jE t)N?pt (E'-hv) N°pt (E')dE
,
PCEtI
(2-34)
S i (EjEv)
N iP t (E-Ev)N^p t (E)
One notable property of E q . (2-30)
is that the fraction of
scattered electrons at energy E is independent of the mean
scattering length and is given by
fraction
scattered
_ Sj(p ^ v )
! + S A ( E jEv)
.
,
(2-35)
-23Equation
(2-34)
indicates that
(E ,hv) -»■ O as E -> h v , that is,
the fraction of scattered electrons becomes
smaller for the
higher energy portions of the EEC's.
C.
Optical Density of States Analysis
The central analysis problem is to obtain the ODS from
the experimentally determined EEC's.
The ODS analysis p r o ­
cedure developed in the present study for determination of
the density of states' below the. Fermi energy is a combination
of methods used by Lapeyre and Kress
(1968) and Eden
(1967)-.
The above methods were extended to include an estimate of the
effects of secondaries
in the EEC's.
The explicit computa­
tional procedure developed for determining the density of
states above the Fermi energy is unique to the present study
but is a direct extension of the implicit computations used
by others
(Berglund and Spicer,
1964).
The following
procedure unambiguously locates the structure
in the ODS but
can only approximately determine t h e .magnitude of the ODS due
to the secondaries
and experimental uncertainties."
The ODS analysis
simplifies
in two limiting cases where
the transport factor becomes
a ( h v ) L (E)
I + a(hv)L(E)
I,
aL >> I
(Case I)
(2-36)
aL,
aL << I
(Case 2)
(2-37)
-24For Case I, the expression for the normalized EDC is given by
AT(E)N°pt(E-h.v)N°pt(E) [1+S. (E-hv)]
N(E-(j),hx)) = ----- ----- 9------------ ------(hv) z EgChv)
AN?pt(E-hv)N®ff (E)[1+S.(E,hv)]"'
= — -------- £-------- ------ (hv)z £2 (hv)
where the functions
off
N_£ (E) .
(2-38)
(2-39)
that vary as E have been summarized as
The limit a (hv) L (E) >> I, implies that the mean
scattering length is large compared to the mean absorption
depth.
With little scattering most of the photoexcited
electrons
should escape and the quantum yield per absorbed
photon should be l a r g e .
Contrary to the above prediction,
yields of molybdenum,
are typically
.01 to
photon for E = 12 eV
the observed quantum
ruthenium and other transition metals
.02 photoemitted electrons/absorbed
(cf. Yu and Spicer,
1967).
The small
values observed for the quantum yield implies the opposite
limit,
a(hv)L(E)
<< I, where most of the electrons are
scattered and consequently do not escape.
In this limit
which is appropriate for transition metals, Eq.
becomes
f
(2-30)
-25-
AT(E)a(hv)L(E)N?pt: (E-hv)N°p t (E) [l + S. (E,hv)]
N (E - <j>,hv)
(hv) Z e 2 (hv)
(2-40)
A N ° p t (E-hv)N®f f (E)[!+Si (E5Ev)]
(2-41)
(hv)n(hv)
where E q . (2-1) and E q . (2-2) have been used and all functions
eff
that vary as E have been summarized as
(E).
Only the
aL << I limit is considered in the following treatment since
the quantum yields of the metals
investigated in the present
study are small for the highest photon e n e r g i e s .
The character of the optical transitions
are found to be
consistent with the nondirect model and the position of
structure in the ODS is determined by simple procedures
provided the contribution of secondaries
is small.
In the
present study there are few secondaries in the EDO's for all
but the highest photon energies
(Chapter IV and V ) .
there are few secondaries S i (E,hv)
If
is small and E q . (2-41)
reduces t o '
A N ^ P t (E-Iiv)Niyf (E)
N(E-<Khv) = — --- (hv)n(hv)--It is apparent from E q . (2-42)
'
'
'
(Z-^Z)
that if the EDO's are plotted
as a function of E - hv the structure in the EDO's due to
N°pt moves by an equal energy increment A(hv)
if
eff
hv -+ hv + A (hv) while that due to N i
remains f i x e d .
This
-26rule,
called the equal increment r u l e , is conveniently tested
for all the E D C 's over a wide range of photon energies by
plotting the energy position of the observed structure versus
photon energy as shown in the hypothetical case presented in
Fig.
4.
The structures
labeled I and 2 in F i g . 4 are
identified with the structure in
and
respectively.
The structure labeled 3 is not consistent with the nondirect
transition model.
A continuous estimate of N ? ^
points
between the structure
is found by including the effects of the secondaries
in the ODS analysis method proposed by Eden
two energies
and ^ 2 •
states at
and hv^ such that
= E-hv^ and
The relative amplitude of the density of initial
and ^
then found
from E q . (2-41)
N^p t (C1) _ N ( E - C ^ h v 1) (hv7 )n(hv? )[l + S. (E,hv )]
_2_ -- L- " ------- — I -----£------ ------- 1------ -—
NYpp(Cg)
Consider
in the optical density of initial states at
Choose E,hv^,
= E-hVg.
(1967).
to be
.
(2-43
N(E-4>,hV2)(hVi)n(hVi)[l+Si(E,hv2)]
In all ge n e r a l i t y , l + Sh(E,hv) must be known before N?p t (C^)
relative to NYp t (Cg) may be determined.
Sh(E,hv)
is a
functional of the optical density of states, N ° pt and N^pt
(Eq.
2-34).
Fortunately,
Eh(E,hv)
is not a strong functional
of the details of N^pt since they determine S^(E,hv)
an integral relation.
through
-27-
E2 = (+ 4 .5 )
— © — © — © — "CD
d)
®
hv (eV)
Fig.
4 .
Hypothetical EDO Structure Plot
-28Since the ODS enters the scattering expression S^(E,hv).
in an insensitive m a n n e r , an approximate ODS will suffice for
a reasonable estimate of the scattering effects.
The zeroth
approximation to the o p t i c a l ■density of initial states,
N i ^ P t (E-Iiv), is found by the technique used by Lapeyre and
Kress. (1968)
results.
and Yu and Spicer
(1967)
to arrive at their final
Their method is based on E q . (2-42)
and the a s sump­
tion that N ^ t t (E) = T ( E ) L ( E ) N ^ P t (E) remains relatively constant
over a small energy interval in the range E = <f>+1.5 eV to
E = 9+3.0 eV.
off
Assuming N^
approximately constant,
zeroth approximation to the shape of N ^ P t (EAhv)
the
is found by.
choosing the shape of the experimental EDO's at 1.5 to 3.0 eV
above the threshold energy.
This
is conveniently done from '
log plots of the EDO's where normalization and hv dependence,
in general can be ignored.
The relative amplitude of various
points in N^Pt (E-Iiv) is made unique within the experimental
error and the error produced by EA(E,hv) by use of E q . (2-41)
(see Krolikowski
(1967)
uniqueness problem).
N ^ P t (Ejhv)
for complete discussion of the
N^ 1 (E) is then obtained by dividing
into the EDO's at various photon energies.
Provided
eff
the resultant effective density of final states, N^
, is
similar for all photon energies,
the technique is assumed to
yield a reasonable approximation to N ^ P t .
In principle the above technique may be used to estimate
-29the density of initial states for E ^ <f> - 11.9 eV since the
photon energy is limited to hv ^ 11.9 eV by the LiF window
transmission.
In practice the threshold function and
scattered electrons severely distort the lower 2 eV portion
of the highest photon energy EDO's
limit of E ^ cf) ~- 9.9 eV.
imposing a more practical
The shape of the highest photon
energy EDO was used as a guide to determine the low energy
cut-off of the density of initial states.
The resultant
cut-off is too slow for the zeroth approximation because of
secondaries but is corrected in the next approximation where
the secondaries are eliminated from the EEC's.
The density of final states above the Fermi e n e r g y ,
, is determined from the density of initial states below
the Fermi energy using the measured optical transition
function and the optical transition strength integral
■2-8).
Previous
investigators
196.4) determined
(Eq.
(e.g. Berglund arid S p i c e r ,
by, an implicit c a l c u l a t i o n .■
was
determined as indicated a b o v e , the form of N ^ t e s t i m a t e d ,
and Eq.
(2-8) ■ computed.
After comparing the resultant.shape
of the calculated optical transition strength function with
the measured value,
(E) was adjusted until satisfactory
agreement was obtained.
A more direct approach is taken in the present study.
Given either N - 0^ t or
IO
I
and the numerical values of the
measured optical transition strength function,
E q . (2-9) is
-30numerically inverted to find
or N°pt respectively.
This
numerical inversion is done' from finite difference approxima­
tion for the. integral expression in Eq.
(2-9),
i.e.
L= ^ = M
( M A E )2 E 2 (MAE) - A Jj N i 0 p t C (L-M-I) A E ] N ° p t [ (L-I) A E ] A E
where the energy scale has been divided into equal increments
of width AE and hv = MAE.
by assuming M = I
The constant A is determined since
and rearranging Eq.
■ ■
(2-44)
gives
' ( A E )2 E 7 (AE)
' A' =
-------------
(2-45)
N o p r (AE) N p A E
where the continuity of the density of states function was
used,
i.e. N^p t (O) = Np = density of states at the Fermi
energy,
Eq.
and all other quantities on the right side of
(2-45)
are known.
With A and N^p t (O) determined,
N o p t C(M-I)AE]
it will be shown that
is determined for all energies where N opt and
the optical transition functions are known.
Then by
induction N^pt is determined for all E < (M-I) E .
The
desired result is obtained by considering only the L = M
term in Eq.
(2-41)
and solving for N^p t , i.e.
-31rP t [ (M-I)AE]
(MAE)
A N c AE
F
e 2 (MAE)
M-I
AAE
%
N?p t [ (L-M-I)AE]N°p t [ (L-I)AE]
(2-46)
L= I
Since the cut-off assumed for N?pt was rather uncertain,
the
zeroth approximation for N^p t , was determined from the above
calculation only for E = 9.9 eV - <j>.
N^pt was extended beyond
this energy by assuming a free electron density of states.
The final estimate of the ODS is obtained from the EDO's
corrected for secondaries.
The contribution of the secon­
daries in the EDO's at' various photon energies
is estimated
by the calculation of E q . (2-38) using the zeroth approxima­
tion to the ODS extended to
|E | ^ 12 eV as described a b o v e , .
The estimated fraction of secondaries was subtracted from the
EDO's to obtain a. set of corrected EDO's.
corrected EDO's available,
With this set of
a slight rearrangement of E q . (2-43)
is used to help define N° p t .
In the present study a combina­
tion of the analysis used by Lapeyre and Kress
(1968) and E q .
(2-43) was used to obtain the final estimate of N?p t .
The
combination of analysis techniques was necessary because of
experimental uncertainties
effects
in the yield and possible collective
in the optical constants for hv = 10 e V . •' The final
estimate for N^ pt was obtained by again inverting the optical
-32transition strength relation
estimate of N ^ t .
(Eq. 2-46) using the final
No further iteration of the ODS analysis'
was deemed reasonable considering the assumption and approxi­
mation involved in the scattering correction.
D.
Summary
An expression for the EDO's of photoemitted electrons
has been obtained including a correction for once scattered
electrons.
The expression has been derived from the nondirect
transition model
(i.e.
conservation of Bloch wave number is
not a good selection rule)
1.
The initial velocity distribution of excited
electrons
2.
and the following assumptions:
is isotropic.
Only inelastic scattering events need be
considered.
3.
The probability of inelastic scattering can be
described in terms of a mean free path which is
a function of the electron energy.
4.
The inelastic scattering is isotropic.
5.
The electron must have a component of its total
crystal momentum P, perpendicular to the surface,
greater than some critical value.
6.
For the transition metals studied a (hv)L ( E ) '<< I
and L ( E 1)ZL(E) .<< I if E'. < E .
-33An implicit assumption of the above model is that manybody effects are negligible.
Quantitative calculations of
the many-body effects in real transition metals are beyond the
scope of present t h e o r y , but have been qualitatively considered
by Phillips
(1965)
and Spicer
(1967).
Many-body effects need
not be invoked to explain the present d a t a .
An analysis procedure for determining the optical density
of states from the nondirect model has been given.
scattering correction is explicitly included,
must be done in two steps:
Since the
the analysis
a zeroth and final approximation.
The analysis procedure is outlined below:
1.
Zeroth approximation
a.
determine
and
b.
calculate N ^ t from Eq.
(2-46) using measured
optical transition strength and N?^-.
2.
Final approximation
a.
using N ^ t and
and extension,
, .
with appropriate cut-off
calculate effect of secondaries
b . subtract secondary from
c . determine
d.
i
EEC's
from corrected EEC's
repeat step l.b. using
.
III.
A.
EXPERIMENTAL PROCEDURES AND EQUIPMENT
Photoemission Measurements
. .
The photoemission measurements of the present study were
done on vapor deposited films produced and maintained under
ultra-high vacuum conditions.
The design of the electronic
and vacuum equipment allowed studies of the E D C 's and the
relative quantum yield to be measured.
designs
The techniques and
are explained in outline form unless they are unique
to the present study.
The reader is referred to the cited,
literature and manufacturer's manuals for more detailed
explanations and characteristics.
I.
Basic Experimental Apparatus
and P r o c e d u r e s . ' A
schematic diagram of the basic apparatus used for measuring
the photoelectric yield and. energy distributions
Fig.
5.
The photodiode consisted of two parts,
is shown in
the cylindrical
collector and the disk-shaped photocathode constructed from
type 304 stainless steel.
The photocathode was highly
polished with a small hole drilled through the center.
The
hole allowed a small fraction of the incident light to fall
on a sodium salicylate
wall of the chamber.
(SS) coated glass window .on the opposite
The SS coated window was cooled to m a i n ­
tain the SS film at room temperature even when the system was
baked.
The photocathode was hinged so that it could be
- 3 5 -
Water
Vacuum
LiF Window
Sodium
Salicylate
coated
Window
Fig.
5 .
Collector
Schematic of Basic Apparatus Used in Photoemission
Experiment.
t
-36- *
withdrawn from the collector by means of a high vacuum linear
motion feedthrough.
In the withdrawn position a gold
(Au)
film was evaporated on the inside of the collector to assure a
surface with an uniform photoelectric work function.
Au was
used to coat the inside of the collector because of its unique
ability of maintaining its photoelectric properties even upon
exposure- to air
(Krolikowski, 1967).
withdrawn from the collector,
the metal to be studied
The photocathode was
and was coated with a film of
(cf. Fig.
6), before it was returned
into the collector for the photoemis.sion measurements.
The entire photodiode assembly was maintained in a high
vacuum chamber. (= 5 x 10
torr)
to insure clean surfaces-
Clean surfaces are needed to minimize scattering of the
photoemitted electrons due to surface contamination.
vacuum LiF window of design reported elsewhere
Spicer,
A high
(Kendig and
1965), was used to transmit the radiation into the
apparatus.
The radiation in the range 2.0 to 12.0 eV was supplied
by a Hinteregger hydrogen discharge
meter normal .incidence monochrometer
lamp dispersed by a one(McPherson Model 225).
Much higher radiation intensities were supplied in the 0.5 to
'6.0 eV range by deuterium, mercury,
and tungsten light sources
dispersed by Bausch and Lomb 33.-88 series m o n o c h r o m e t e r s .
all energies used,
the optical band was less than 0.1 eV.
For
-372.
Vacuum Equipment and P r o c e d u r e s .
The only vacuum
equipment and procedures described in detail will be those
used in conjunction with the photoemission studies.
The
equipment and procedures used in the optical measurements
were similar to that used for photoemission Studies unless
otherwise, noted..
The stainless steel chamber housing the experimental
apparatus was built from standard high vacuum fixtures and
from a few custom modified.fixtures,
6.
and is displayed in F i g .
All internal apparatus was constructed from low vapor
pressure refractory materials
such as non-magnetic 304 series
stainless
The selection of materials
steel and O F H C -Cu.
allowed the capability of baking the system to 200° C .
.
Baking
at.elevated temperatures was necessary for the outgassing
needed to achieve the desired high vacuum.
The materials
exposed to the high vacuum were carefully cleaned and handled
according to standard high vacuum cleaning procedures
(Rosebury, 1965).
The high vacuum, system was
free Vacsorb
(Variari) pump.
initially evacuated by ah oil
The pressure was further reduced
with the aid of an 8 1/sec VacIon
(Varian) p u m p .
High vacuum
pressures were achieved and maintained by an 800 1/sec OrbIon
(Norton) p u m p .
The pressure of the high vacuum system,
as
measured by Bayard-Alpert ionization g a u g e , was about 5 x 1 0
Linear Motion FeedthroughViewport—
[j
p
I
I-^i-E le c tric a l
Feedthrough
Au
>
Photomultiplier Tube
Ionization^(^__^
X
LiFWindow
o
•/:^ v / " v y x y h v
Sodium
Salicylate
Gauge
To 8 0 0 l/s e c
O rbIon
Pump
!%
r
'\
8 1/sec
VacIonB
VacSorb
Pump
High Vacuum
e -G u n -
Fig.
6 .
'7=-
Valve
Schematic of Vacuum Chamber for Photoemission
Studies.
-39torr after baking the entire system at -200°C for 12 to 48
hours.
Since the r e f lectometer had detectors which could not
be baked above 7 5 0C , only those portions remote from the
detectors were baked.at ZOO0C .
3.
Sample P r e p a r a t i o n .
a thick metal film
vacuum.
The photocathode was coated with
1000 A) by vapor deposition under high
..The metal sample was vaporized by a water cooled
electron beam deposition gun
chamber as seen in F i g . 6.
(Varian) near the bottom of the
When the electron gun and the
metal sample had been carefully outgassed,
could be produced at 5 x 10
-9
torr.
evaporated films
The total evaporation
time could not exceed several minutes because radiation from
the molten refractory metals would heat and outgas the
surroundings.
These metal films were contiguous with the
polished stainless steel photocathode
smooth.
and appeared visibly
Subsequent x-ray powder patterns of samples scraped
from the photocathode were characteristic of the p o l y c r y s t a l ­
line pure metal.
4.
Energy Distribution Curve M e a s u r e m e n t .
The kinetic
energy distribution of the photoemitted electrons was measured
by the retarding potential method.
A potential difference that
varied linearly wit h time was applied between the photocathode
and the collector while the radiation incident oh the p h o t o ­
cathode was maintained at a constant flux level and w a v e l e n g t h ,
-40Fig.
5.
An operational amplifier differentiator was used to
time differentiate the instantaneous
meter monitoring the photocurrent
signal from the ele c t r o ­
(Kress and L a p e y r e , 1968).
The resultant signal is proportional to the number of
electrons which have the same kinetic energy as the instan­
taneous potential difference across the photodiode,
i.e.
proportional to the kinetic energy distribution of the
photoelectrons.
The physical equipment and electronic
details of this measurement have been published
(Kress and
L a p e y r e , 1968).
A small contact potential was observed between the
photocathode and collector
(see Fig.
9).
This small contact
potential was known in both sign and magnitude since the
work function of the collector
photocathode was measured
(Au)
(Decker,
is known and that of the
1954).
The effect of the
contact potential was eliminated from the m e a s u r e m e n t s .
Experimental errors and uncertainties in the EDC m e a s u r e ­
ment by the retarding potential method have been discussed in
detail by other
(cf. Eden,
1967 and K r o l i k o w s k i , 1967).
Their
discussions are applicable to the present study if the error's
and distortions due to the operational amplifier d i fferentia­
tor designed and built for the present study are included
(Kress and L a p e y r e , 1968).
The overall effects of these
errors and distortions produce an experimental uncertainty of
-41structure in the EDC of about ±0.1 eV.
The position of the
Fermi energy is determined within similar experimental limits.
5.
Quantum Y i e l d .
The absolute quantum yield,
defined
as the number of photoemitted electrons per absorbed p h o t o n ,
was measured by first determining the relative quantum yield
per absorbed p h o t o n .
The relative quantum yield can be
determined if the photocurrent,
normalized to the photon f l u x ,
and the reflectance are known as a function of photon energy.
The total photocurrent is easily measured at a given flux
level by maintaining a +10 volt potential between the
collector and photocathode while monitoring the photocurrent
with an electrometer.
The p h o t o c u r r e n t , I
is given by
■ = &(hv) Y (hv)
p.
l-R(hv)
where
(3-1)
I (hv) is the incident photon flux, Y (hv) is the absolute
quantum yield,
absorbed.
I - R (hv) represents the fraction of the flux
The more difficult relative photon flux m e a s u r e ­
ment was determined from the small fraction of the photon
flux that passed through the photocathode and caused flu o r e s ­
cence of the SS coated window opposite the LiF entrance
window as shown in F i g . 5.
A photomultiplier tube
(EMI 9514)
outside the high vacuum chamber was used to monitor the
fluorescence from the S S .
Assuming the relative fluorescent
quantum efficiency of SS is constant,
the current from the
-42photomultiplier is
Ip = C JtQiv)
where C is the effective gain of the detection system.
(3-2)
The
signal from the photomultiplier tube was electronically
divided into the photocurrent signal in a continuous manner
by means' of an operational amplifier voltage divider
(Philbrick-Nexus).
The output of the divider is proportional
to the yield per incident photon.
(3-3)
where C
is a constant summarizing all the effective gain
factors.
The divided signal described by E q . (3-3) was recorded on
the Y axis of an XY- recorder as the photon energy was c o n t i n u ­
ously changed with time..
A signal proportional to the
instantaneous photon energy, h v , was simultaneously recorded
on the X axis
photon
(Kress, 1969).
The recorded yield per incident
(Eq. 3-3) was corrected by the measured reflectance to
obtain the relative quantum yield per absorbed photon.
The
absolute yield was determined at 10.2 eV from the absolute
light flux
(NO c e l l , M e l p a r ) , the LiF window transmission,
and
the absolute photocurrent.
The above method of measuring the relative quantum yield
-43of a metal relative to the response of a SS film inside a high
vacuum chamber appears to be unique to this work and will be
considered further.
There are two possible reported sources
of error that result from using the relative fluorescent
quantum efficiency of SS for the measurement of the light flux
Recent studies
(Samson,
1967)
indicate that a decrease in the
efficiency of SS occurs in the wavelength region 1600 X to
O
1000 A (- 8 to 12 e V ) . This decrease is approximately 10 to
20 p e r c e n t .
In addition to the relative efficiency problem,
some reports of time dependence of the relative efficiency of
SS have appeared in the literature and were noted in this
laboratory when SS was maintained in a vacuum produced by a
diffusion p u m p .
Contrary to these observations no aging was
found in the present study
et al,
(see below)
and by others
(Allison
1964) when SS was maintained in an oil f r e e , vacuum
environment at room temperature.
The aging and flatness of the SS coating used in the Mo
study was indirectly checked by comparing the yield of Au
obtained by the above method with that obtained by means of a
detector that- was calibrated at several points w i t h a vacuum
thermopile
(Krolikowski, 1967).
Before comparing with the
Krolikowski results,
the relative yield per incident photon
<
measured by the above technique was corrected with the
published reflectance of Au
(Canfield and H a s s , 1965)
and
-44arbitrarily normalized.
The results
are displayed in Fig.
7.
The agreement over the measured range of photon energies is
good.
The greatest experimental uncertainty of the relative
quantum yield arises from the uncertainty in the relative
fluorescent quantum efficiency of S S .
Combining the estimate
with the errors introduced by the electronic voltage divider,
it is estimated that the absolute values of the yield reported
may be in error by as much as 25% for energies remote from the
one calibrated point
(10.2 e V ) .
energy range, however,
Within a small
(I or 2 eV)
the relative values of the yield have
only several percent relative uncertainty.
B.
Optical Measurements
The optical "constants" of a metal may be determined
when the reflectance at near normal incidence is known over a
wide range of photon energies
and the reflectance at some
angle other than normal incidence is known over a limited
energy range
(Appendix B ) .
It is important that the surface
of the metal remains clean during optical measurements since
small amounts of contaminants have noticeable effects on the
optical functions in the vacuum ultraviolet region
(Beaglehole,
1965).
The evaporated film, therefore, must be
prepared and the reflectance measurement must be done in a
high vacuum e n v i r o n m e n t .
A refhectometer which could meet
“ 4 5 _
G x IO
4 x IO
Quantum
Yiaid
(electrons/absorbed photon)
2 x IO"2
2 x IO
----- Krolikowski
----- Present Study
hv (eV)
Fig.
7 .
Q u a n t u m Yield of Gold Compared to That Taken by
Krolikowski.
-46t h e s e 'requirements was constructed, and is briefly described
below.
I.
Reflectometer.
The reflectance measurements in this
study were taken by means of a high vacuum r e f lectometer
designed and built mainly by L a p e y r e , H o l v e r s o n , and Stensland
(1969).
A schematic diagram of the reflectometer and vacuum
system used in the present study is presented in F i g . 8.
smaller 250 1/sec VacIon
The
(Varian) pump produced base pressures
and evaporation pressures very similar to the 800 1/sec
OrbIon
(Norton) pump used in the photoemission studies.
The reflectometer was designed to allow the reflectance
at arbitrary angles between 12°. and 70° to be observed as
described below.
After coating the substrate
(a clean
microscope slide) by means of vapor deposition of the
appropriate metal,
the substrate was rotated to a horizontal
position as shown by the dashed line in Fig.
8.
The detector
was then moved into the light and rotated until the maximum
obtainable signal was found
(dashed c i r c l e ) .
The substrate
was moved to the desired angular position and the detector .
was rotated until its signal was maximized again.
The
position of the detector indicated the angle of reflectance
and the ratio of the above two signals was recorded as the
reflectance.
Two detectors were used to span the complete energy
•-47-
SUBSTRATE
VIEWPORT
r-
DETECTOR
TO 2 5 0 L /s
ION PUMP
—
Fig.
8 .
* c - G U N
Schematic of Reflectometer Used for Reflectance
Measurements.
-48range of i n t e r e s t .
multiplier
From 2.0 eV and 12.0 eV a 1P28A p h o t o ­
(PM) with a thin SS coating was mounted inside the
r e f le c t o m e t e r .
In the ultraviolet region,
the fluorescence
of the SS film provided the detectable signal for the PM.
Below 6.0 eV the envelope of the PM and the thin SS film was
semi-transparent and allowed the incident radiation to fall
directly on the photocathode of the PM.
Since the threshold energy for the PM is approximately
2.0 eV, another detector was needed to extend the optical
measurements below 2.0 e.V.
(PbS)
(I x 5 mm)
.the reflectance.
to.3.0 eV.
An Eastman Kodak lead sulfide
detector was chosen to extend the range of
The PbS detector was sensitive from 0.5 eV
PbS has a high vapor pressure and was kept from
evaporating by means of a glass coverplate sealed by V a r i a n ts
Torr Seal.
The Torf Seal raised the base pressure in the
-Q
r e f lectometer to -2 x 10"
torr.
base pressure,
In spite of the degraded
the data obtained in the near infrared region
reproduced the high vacuum data where they overlapped indica­
ting a lack of contamination.
Errors and uncertainties
in the reflectance measurement
at normal incidence were checked by measuring the reflectance
of Cu between 2.0. eV and 12.0 eV.
These observations
reproduced the’published Cu data within ±2.5%.
value of the reported reflectance
The absolute
is thus not expected to be
-49greater than ±2.5%.
The relative values of the reflectance
are considerably better.
the uncertainties
There is no simple relation between
in the reflectance and the optical constants
(Beaglehole, 1965).
C.
Data Reproducibility
Reproducibility of the structure and magnitude of all the
photoemission and optical data was the minimum criterion used
to determine a "good" set of data.
The metal films were
evaporated with the experimental chamber coupled to the light
source and all the electronic measuring and recording a p p a r a ­
tus were operating for both the photoemission and optical
studies.
Within one minute of completing a film,
measurements were t a k e n .
the initial
The first measurement was retaken
periodically as a check on any aging effects.
In general,
there were no observable aging effects over a four to eight
hour period when the base pressure was in ^ 5 x 10
range and deposition occurred in the 10
-9
films were maintained for longer periods,
-10
torr range.
torr
If the
small changes would
appear indicating contamination of the metal surface by the
residual gasses in the vacuum c h a m b e r .
The., observed aging effects, were small and did not change
the major features of the EDC or the quantum yield.
assertion is illustrated by the EDO's in Fig.
directly from the experimental charts.
This
9, plotted .
Mo film IV was about.
-50-
— Mo 12
N (E-<£, hy)
— Mo 2
RETARDING POTENTIAL (volts)
Fig. 9 .
Direct Tracing of Experimental EnergyDistribution Curves for hv = 10.2 eV
from Experimental Chart.
-5124 hours old when the illustrated EDC was t a k e n .
Film V was
evaporated over Film IV and the EDC immediately retaken with
all the same experimental parameters except for an 8% increase
in lamp current needed to stabilize the lamp.
It is apparent
that the fraction of high kinetic energy electrons
is
decreased in the EDC of film IV as compared to that of Film V .
Since the width of the EDC from Film V is 0.3 volt less than
that of Film IV, the photoelectric work function is less.
If
the area of the EDC from Film V is decreased by 8% to c o m p e n ­
sate for the change in the lamp intensity,
the difference in
the areas of the two EDC's is a direct measure of the d i f f e r ­
ence in the quantum yield.
above results
The difference is very small.
The
are typical of the aging o b s e r v e d .
The reproducibility of the measured work function was not
good in the initial measurements.
It had been assumed that
the yield and threshold function measurements were less
sensitive to surface contamination than the EDC's measurement.
Hence EDC's were taken,
then the threshold and yield data.
When the order of data taking was changed and the threshold
data completed first,
the threshold measurement became very
reproducible.
Reproducibility of the reflectance data was relatively
insensitive to evaporation pressures
Evaporations
and aging effects.
in the 10 ^ torr range would agree well with
-52those done in the 10
-9
torn r a n g e .
The overall magnitude of
the entire reflectance curve obtained from individual films
was shifted by I to 2% from the average magnitude.
These
shifts did not correlate with the evaporation or base
pressures.
Although the magnitude of the various samples
had some film dependence,
the structure observed in the
reflectance from different films reproduced on all the
observations.
It was concluded that these shifts
in
magnitude were due to the nature of the metal film itself.
IV.
A.
MOLYBDENUM -EXPERIMENTAL RESULTS
Optical Measurements
In the present c h a p t e r , the results of optical and
p h o t o emission measurements of the metal molybdenum
are presented.
(Mo)
The experimental procedure used to obtain the
following measurement is given in Chapter III.
The definition
of the functions with their theoretical development is given
in Chapter II.
Discussions relevant to the analysis p r o c e ­
dures are given here. - Interpretation of the results is mainly
reserved for Chapter V I .
I.
R e f l e c t a n c e ..
It is shown in the theory section and
in Appendix A that the optical constants may be derived from
the reflectance at near normal incidence and a segment of the
reflectance at another a n g l e .
Such measurements were taken
between 0.5 eV and 11.9 eV on four vapor deposited films of
Mo as described in Chapter III,
(solid l i n e ) .
and are summarized in Fig.
The best evaporation pressure was =5 x 10
torr with a base pressure of <5 x 10 ^
torr.
Jarellash
(99.99% p u r e ) , and Materials Research Corporation
(99.995% pure,
carbon).
-9
The films were
produced from samples obtained from two s o u r c e s :
Company
10
in particular less than 15 parts per million
The reflectance from a mechanically polished bulk
sample purchased from Sylvania Corporation
(99.5%) was also
obtained at near normal incidence between 2.0 eV and 14.0 e'V.
“54“
<
0.3 -
.......KIRLLOVA1BOLOTIN, 8 MAYESKh
------WALDRON 8 JUENKER
------LE BLANC, PARREL,8 JUENKER
------PRESENT STUDY (high vocuum)
^ - P R E S E N T STUDY (low vacuum)
Fig.
io.
Summary of Reflectance Measurements
for Molybdenum.
The reflectance of Mo has been measured by others.
Kirillova et al.
samples
(1967) used mechanically polished bulk
exposed to the atmosphere to' find the reflectance
between 0.05 eV and 12.0 eV.
Waldron et al.
(1964) measured
a heat cleaned bulk sample in a high vacuum glass tube
between 2.0 eV and 5.0 eV.
LeBlanc et al.
(1964) used a
heat cleaned bulk sample in a high vacuum stainless steel
chamber and photoemission technique to obtain the reflectance
between 6.0 eV and 23.0 eV.
These other studies agree well
with each other but differ from the present study as seen
in Fig.
10.
The disagreement requires some discussion.
The data of Waldron et al.
under high vacuum conditions
and LeBlanc et al. were taken
(=10 ^ to 10
torr).
Their
sample consisted of a bulk sheet of Mo which was cleaned
before placing it in. the high vacuum.
conditions,
Under high vacuum
the Mo was flashed to a very high temperature
(=2200°K) immediately before taking reflectance measurements.
The assumption was made that this severe heating would produce
a clean surface of Mo since most materials evaporate more
rapidly than Mo at these elevated temperatures.
This is not
true of carbon, which has a vapor pressure versus temperature
dependence almost identical wit h Mo
(Rosebury,
1965).
Some
of the more recent literature on Auger electron emission from
Mo
(Vance,
1967)
indicates
that carbon contamination of the
-56surface of Mo is increased by diffusion of carbon from the bulk
material when a bulk sample is heated to high temperatures
a high vacuum environment.
in
The above observations indicate
the optical data obtained from heat treated bulk samples of Mo
should be treated with caution.
To check the data reported by Kirillova et al, the
reflectance of a mechanically polished ( A l ^ polishing grit)
bulk sample was also measured in the present study in a low
vacuum re f lectometer designed and built by others
(Chor, 1967).
The results of the low vacuum reflectance measurement are seen
in Fig.
10.
This measurement did not reproduce the data of
Kirillova et a l . or that of Waldron and L e B l a n c .
It did
reproduce all the structure of the high vacuum data taken in
the present study with a small attenuation in the general
magnitude of the reflectance.
Since unusual carbon c ontamina­
tion of sample prepared by mechanical polishing is not expected,
it is difficult to understand the differences between these
observations and those of Kirillova et al.
The agreement between reflectance measured from the airexposed mechanically polished bulk sample and the evaporated
films formed under high vacuum conditions, was taken as
confirmation that the high vacuum results of the present work,
are the best measurements of the reflectance
0.5 eV < hv < 12.0 eV at this time.
of
Mo for
This conclusion is
.
-57supported by the observed similar magnitudes of the reflectance
of the high vacuum.Mo data of the current study and that of W
(LeBlanc'et a l , 1964).
The optical properties
are expected to
be similar since the electronic structure of Mo and W are similar.
The optical constants of Mo have been extended to 23.0 eV
by use of LeBlanc's reflectance data.
His reflectance curve
was multiplied by the constant needed to join his curve
smoothly to the reflectance measured in this work at about
12.eV.
The results are given in Fig.
11.
This procedure is
justified by the empirical observation that the addition of
LeBlanc's data beyond 12.0 eV does not significantly change
the calculated optical constants
of Mo below 12.0 eV. compared
with the results obtained by cutting the measured reflectance
off at 12.0 eV.
No interpretation made in this study will
depend on the optical functions beyond 12.0 eV.
The optical constants of Mo were calculated by KramersKronig's
analysis of the reflectance.
The analysis can be
completed only if a high photon energy extrapolation of the
reflectance is assumed.
The best fit, as explained in
Appendix A, for the measured reflectance at 50° angle of
incidence was obtained with .an extrapolation of the form
R = (1/E)^
for E > 23.0 eV.
Given this extrapolation
n ( h v ) a n d k(hv) were found and all the other optical constants
calculated from n(hv)
and k(hv).
" 5 ^ ~
R (hv)
=
12
- i = 5O
.
11.
IO 12 14
hv (eV)
16
18
20 22
Reflectance of Molybdenum './here
i Is the Angle of Incidence.
(hv),€2(hv)
£
8
8
Fig.
12.
IO 12 14
hv (eV)
16
18
2 0 22
Dielectric Function of Molybdenum.
-59The high energy extrapolation introduces some error into
the optical constants since the exact high energy reflectance
is not determined.
The error introduced by the high energy
extrapolation produces some uncertainty in both the magnitude
and energy position of structure in the optical constants.
The energy position of structure in the optical constants was
found to be insensitive to large changes of the exponent in the
high energy extrapolation.
' The magnitude of the optical
constants was more sensitive to the value of the exponent.
The magnitude of the present calculation for n (hv) and k (hv)
are compared with values measured by techniques
depend on a high energy extrapolation in Table
that did not
I.
Considering
TABLE I
Energy
Optical constant
n
2.15 eV
1.00 eV
k
Study
’
• 3.1
2.6
(Present study)
3.2
3.4
(Juenker et al, 1968)
3.6
3.3
(Summer, 1934)
4.4
3.6
3.0
4.1
(Waldron and J u e n k e r ,
1964)
(Present study)
3.0
4.8
(Kirillova et al, 1967^
-60the magnitude differences observed in the reflectance of Mo
compared with the reflectance observed in the studies s u m m a ­
rized in Table
2.
I (cf. Fig.
10),
the agreement is reasonable.
Dielectric C o n s t a n t .
imaginary part,
Cg(^v),
The real part,
e^(hv),
and the
of the complex dielectric constant or
response function is presented in Fig.
12.
If the electrons
in the metal had free electron character only,
would
monotonically fall to zero while C 1 would monotonically rise
to one with increasing energy
from the rapid fall in
(cf. Eq.
(2-10)).
are centered at 1.8 eV and 3.8 eV.
The marked deviation of
from the free e l e c t r o n - l i k e ,
monotonic rise towards unity begins early
ceases near 5 to 6 eV.
Deviations
The behavior of
transitions below 6 eV.
(I to 2 eV) and
indicates numerous
.
...
The location of the beginning of the interband tran s i ­
tions in Mo is not possible from the present studies of e(hv)
since these measurements extend only to 0.5 eV.
however,
Lenham
(1965)
studied the infrared behavior of the optical function
2nk/A and found the interband threshold to be 0.44 eV.
Furthermore,
Lenham found that optical constants of Mo were
not free electron-like within the limits of his measurements
(=0.1 e V ) .
Kirillova et al
optical functions.
(1967)
also studied the infrared
They found that the onset of the interband
transitions began at 0.17 eV.
In either case, the optical
-61functions of Mo,
transition metals
as the optical functions of most of the other
(Lenham,
1965) , do not have free electron
character even in the near infrared region.
The lack of free electron-like behavior for the optical
functions of Mo in the range 0.5 eV to 2.0 eV was verified
in the present s t u d y .
Attempts were made to separate the
free and bound electron contributions
to the dielectric
function as has been done with the noble metals Ag and Cu
(Ehrenreich and Philipp,
1962) .
That is, the free electron
expression for e^., E q . (2-11), was fitted to the experimentally
determined e in the range 0.5 to 2.0 eV by considering t , the
relaxation t i m e ,. as a p a r a m e t e r ..
It was f o u n d , h o w e v e r , that
T could not be considered a constant,
i.e.
t
had energy
dependence.
3.
Loss F u n c t i o n s .
The energy of collective volume and
surface excitation is determined from the peaks
and surface loss functions r e s p e c t i v e l y s i n c e
of the v o l u m e ■
the optical
constants satisfy the conditions presented in detail in
Chapter II. ■ A peak in the volume loss function,
Im 1/e
(where
Im denotes "imaginary part of"), occurs at 10.8 eV while that
of the surface loss function,
as seen in F i g . 13.
Im I / (e + 1 ) , occurs at 9.9 eV
The energy of volume plasmon resonance is
in rough agreement with that found by LeBlanc at 10.0 eV. This
energy is also in reasonable agreement with characteristic.
—62—
Im I/e
6
8
IO 12
14
16
18 20 22
hv (eV)
Loss Functions for "oIybdonum
2 x IO
cm-l
2
4
6
IO
12
14 16
18
20 22
hv(eV)
Fig.
14.
Absorption Coefficient cx for
Tiolybdenumu
-63energy loss measurements of fast electrons reflected from Mo.
Haworth .(1935,1936)
found, peaks
in the characteristic energy
loss spectrum of Mo at 10.6 eV and 22.0 eV.
on the other hand,
Kleiner
(1954),
found them at 12.0 eV and 24.5 eV.
The
lower value of Haworth is in good agreement with this study
and that of Kleiner in rough a g r e e m e n t .
Neither Haworth nor
Kleiner resolved two p e a k s , one due to the volume and the
other due to the surface e x c i t a t i o n .
In general,
collective
oscillations are determined with better resolution from
optical experiments
than from characteristic energy loss
studied for low and m e dium energy
(Arakawa et al,
(=25 eV) excitation
196 5) .
Assuming six free electrons per atom for Mo,
electron model gives hv^ = 23 eV.
the free
L e B l a n c 's optical investi­
gation beyond the energy range of this study is consistent
with a classical, plasma resonance in this energy region.
4.
Alpha,
(hv)n(hv),
(hv)^
(hv). The optical
■ 2",
(hv)
Ev(Hv) are displayed
functions a (hv) and
(hv)n (hv) and
in Fig.
15 r e s p e c t i v e l y .
cient,
14 and Fig.
a(hv) , has a relative minimum for hv = 11.3 eV.
Consequently,
the mean excitation depth for electrons
■maximum at this energy.
(hv)
The absorption c o effi­
(hv) have
small differences
The optical functions
large differences
in s h a p e .
is a
(hv)n(hv)
and
in magnitude, but only
These functions will be used in
the analysis of the photoemission data.
-64-
(hv) n(hv)
—
(hv) e (hv)
J—
4
z(A3) (AMPz(Ml)
—
i__I__ i__ I___L_i... I__ I___ L
6
8
IO
12
14
16
IO
2 0 22
hv(eV)
Optical Functions (hv)n(hv) ar
(hv)
2 (h v ) for Molybdenum.
§
in
Cl
o
O
CD
CD
in
(U
3
OC
E
13
CO
4
6
8
—U--1
IO 12
14
16
18
2 0 22
hv (e\Z)
Fig.
• Interband N t- , Surface Plasma N
and Volume Plasma N y p Sum Rules
for Molybdenum.
Sp
-655.
Sum R u l e s .
(Ngp), and volume plasmon
The interband
(Njb) / surface plasmon
(Nyp ) sum rules calculated from
E q . (2-12), E q . (2-13) , and E q . (2-14) r e s p e c t i v e l y , are
displayed in Fig.
16.
Since Mo has six electrons
(4d^5s"*") in
the outer s- and d- s h e l l s , it is apparent that the interb.and
sum rule has not reached its expected limiting value of six
electrons even in the range of 23 eV.
for a transition metal.
for palladium
(Pd)
approximately 50 eV
This is not atypical
For e x a m p l e , the interband sum rule
is not saturated at 11 electrons until
(Robin,
1965).
..
The number of electrons
participating in the collective volume and surface plasmon
excitation is very small below their measured energy values of
10.8 eV and 9.9 eV respectively.
photoemission data was taken
Over the range in which
(hv ^ 11.9 e V ) , interband t r a n s i ­
tions, not collective excitation,
are the dominant optical
excitation process.
.B .
Photoemission Measurement
Photoemission measurements of vapor deposited films of Mo
on a stainless steel substrate were made under high vacuum
conditions. • The- best deposition pressure was
with a base pressure of 6 x 10
after the deposition.
”10
7x10
-9
torr
torr achieved a few minutes
A total of twelve films produced from
samples supplied by the two sources indicated in Section A,
Part I were investigated.
Excellent reproducibility of the
-
66
-
data was found among all the measurements taken on films
deposited at 10 ^ torr or better.
I.
Yield.
The quantum' yield
(electrons/absorbed photon)
as a function of photon energy for Mo was measured by the
procedures and equipment explained in Chapter III and is
presented in F i g . 17 along with the Fowler
determine the work function as 4.3 eV.
(1931) plot used to
The photoelectric work
function is in excellent agreement with other recent m e a s u r e ­
ments of the work function of polycrystalline Mo "as summarized
by Vance .(1967).
The quantum yield shows no strong structure.
The rapid increase in the yield tapers off near 10.0 eV
correlating well with the appearance of the last observed peak
in the E D C s
(see below) .
A significant drop in the yield is
centered at approximately 11.0 eV.
peak in the volume loss function
This correlates with the
(10.8 eV) and the relative
minimum in the absorption coefficient
minimum in a(hv)
excitation depth,
decreases.
(11.3 e V ) .
The relative
implies a relative maximum in the p h o t o ­
t h e r e f o r e , it is reasonable that the yield
The decrease in the yield at approximately 11.0 eV
is consistent with the optical constants.
The value of the observed quantum yield at the highest
photon energies available in the present study
was about
.02 electrons/absorbed photon.
(hv ^ 11.9 eV)
This small value
.
for the yield indicates that the limit ah << I is appropriate
Quantum Yield (electron/absorbed photon)
— 6 7 "
4.OqV 4.58V 5.OcV
hv (eV)
F ig ,
I?
Quantum Yield for Molybdenum
for Mo,
according to the discussion given in Chapter II.
2.
Energy Distribution C u r v e s .
The energy distribution
curves for photoemitted electrons frqm vapor deposited films
of Mo have been obtained as a function of photon energy.
Representative samples of the EEC's normalized to the quantum
yield for Mo are shown in F i g . 18 plotted as a function of
E - <J>•
The structures
kinetic energies.
in these.EEC's.do not have fixed
Since there is no strong structure which
is fixed with E - <j> (or equivalently E) ,• there is no such
structure in
‘ (E).
Arbitrarily normalized EEC's of Mo are presented in Fig,
19 plotted as a function of E-hv.
There are three distinct
peaks, seen in the EEC's labeled I, 2, and 3 which have fixed
positions versus E - h v .
In addition there is a sudden
deterioration in the high energy edge of the EEC's between
hv = 10.0 eV and 11.0 eV.
This effect correlates
well with
the onset of the volume plasmon at 10.8 eV and maximum in the
escape depth implied by the minimum in a(hv)
at 11.3 eV.
The
fractional change in the high kinetic energy electrons lost
between 10 eV and 11 eV is not great and alternately could be
a result of a decrease in L(E)
or even an indication of direct
transition character in this energy region.
The energy dependence of peaks
range of photon energies
I, 2, and 3 over a wide
is summarized in the structure plot
6x!0-3
e II.8
N ( E - 0 , hv)
electrons/pho1on-eV
SxIO "3
hv = 5 .OeV
E -0
Pig.
18.
(eV)
Normalized Energy Distribution Curves for Molybdenum Plotted Versus F - 0 .
-70-
hv = no cv
E - hi/
Fig.
19.
(eV)
Arbitrarily Normalized Energy
Distribution Curves of Molybdenum
Plotted Versus E - hv.
-71presented in Fig.
peaks
20.
It is clear from Fig.
20 that the
I, 2, and 3 have E - hv dependence over a wide range of
photon energies.
These peaks may therefore be associated with
peaks in the density of initial states located at -0.5 eV,
-1.6 eV,
and -3.9 eV r e s p e c t i v e l y . '
A set of normalized EEC's multiplied by the transport
factor in the limit ah << I, i .e.. (hv)n(hv),
Fig.
21.
is displayed in
The consistent overall magnitude or overlap of this
set of curves indicates
that
(hv) n (hv)N (E- cj),hv) may be well
described by a product of two functions of the form
N ? P t(E-hv ) N ® £ £ (E), where N ^ ££ is not rapidly varying with
energy.
3.
Optical Density of States for M o l y b d e n u m .
To obtain
the zeroth approximation of the ODS needed to complete the
scattering correction for secondaries,
a zeroth approximation
to the density of initial states, N ? £ £ (E-hv) was determined
as indicated in Chapter II.
This zeroth approximation to the
optical density of states was then divided into the EEC's at
various photon energies to obtain the effective density of
eff
final states .N^
(E) . k composite of the shapes of the
off
various
(E) obtained is presented in F i g . 22.
The e f f e c ­
tive density of states for photon energies not shown, will
fall within the bounds defined by the curves shown.
The
superposition of the effective density of final states as
•
(eV)
Energy of Structure
5.0
6.0
7.0
8.0
9.0
10.0
11.0
hv (eV)
Fig.
20
Structure Plot for
Molybdenum.
12.0
N(E-HvJizz) x (hv)n(hv) (arbitrary units)
Fig.
M.SeV,
21.
Normalized Energy Distribution Curves Multiplied by (hy)n(hv) of
Molybdenum Verses E - hv.
(arbitrary units)
Ne^ (E)
hz/= 8eV
Fig.
22.
Nf0x" (E) for Various Photon Energies.
-
75
seen at various photon energies
-
indicates that the EDO's can
be described as a product of a function that varies as E and
one that varies as E - hv.
The zeroth approximation to N ^ P t (E-Iiv) and the optical
transition strength,
(hv)
2
££ C^lv) > were used to determine the
zeroth approximation to N ^ P t (Chapter I I ) .
The zeroth
approximation to N ^ P t was smoothed so that only the general
envelope was considered significant.
presented in Fig.
N ^ P t and N ^ P t are
23, where the dashed line indicates the
assumed low energy cut-off and high energy free electron
extension.
The calculation for the density of final states,
knowledge of
(hv) for small photon energies.
assumes
In the present
study the reflectance and. consequently the optical constants,
were not determined for hv ^ 0.5 eV.
for
(hv)
Without accurate results
at the lower photon energies,
the optical density
of final states and the constant. A, cannot be calculated
(cf. E q . 2-45).
This difficulty was circumvented by choosing
a constant average value for N ^ P t (E) which produced approxi­
mately as many empty states
(six for Mo and four for Ru) as
required to fill the. empty d-band in the next several e V .
The
states near the Fermi energy may be arbitrarily chosen since
the exact magnitude of the states
in this small region was
found by explicit computer calculations,
as intuitively
-76-
-IO - 8 - 6 - 4 - 2
0
2
4
6
8
10
E (eV)
Fig.
23.
Zeroth Approximation to the
Optical Density of States
Used to Estimate the Scattered
Electron Contribution to the
Energy Distribution Curves of
Molybdenum.
-77expected,. to have a negligible effect on the overall shape of
the density of final states.
The. fraction of once scattered secondary electrons in
the EDO's was estimated from E q . (2-35) and the zeroth
approximation to
and
.
.The secondaries were then
subtracted from the measured EDO's.
The results for Mo are
summarized by two EDO's calculated at hv = 8.0 eV and 11.0 eV
given in F i g . 24 and F i g . 25 respectively.
As expected,
the
high energy edge of the EDO was only slightly effected while
the low energy portion acquired a large correction.
fraction of secondaries
significant in F i g . 25.
electrons
The
i s .negligibly small in F i g . 24, but is
Since there are more energetic
for hv = 11.0 eV, it is reasonable that the c o rrec­
tion due to secondary scattered electrons should become larger
with increasing photon energy.
It was found empirically that the scattering corrections
were very insensitive to the details of
.
If structure
was included in N ^ t , it produced only very minor structure in
the corrected r e s u l t s .
The magnitude of the scattering
corrections' was somewhat more sensitive to the relative
magnitude of N ^ t compared to N ^ P t .
This ratio was chosen as
indicated above.
The final approximation to N?^^(E-hv) was then Obtained
from the corrected E D C s
as explained in Chapter II.
With
6 Xio
N (E- <£>,8)
electrons / photon-eV
Uv=
8.0eV
— measured
-- corrected
••• scattered
4 x 10
2 x IO
0
1
2
3
4
5
6
E- <t> (eV)
Fig.
24.
Measured, Corrected, and Scattered
Energy Distribution Curves of
Photoemitted Electrons for Tm
Molybdenum at hv = 8.0 eV.
-79-
6x10'
— measured
— corrected
••• scattered
Fig.
25.
Measured, Corrected, and Scattered
Energy Distribution Curves of Photoemitted
Electrons from Molybdenum at hv = 11.0 eV.
-80this final .approximation to N ^ P t (E-Iiv), the optical, transition
strength integral was inverted again to arrive at the final
approximation to N ^ P t (E).
These are displayed in Fig.
26.
No
further iterations of the ODS analysis technique deemed
reasonable considering the assumptions and approximation
involved in the scattering correction.
The reader is cautioned that the numerical inversion of •
the optical transition strength function for N ^ P t was done in
0.2 eV increments,
hence the resolution could certainly not
be better than 0.2 eV.
Recalculating the optical transition
strength function with the density of state shown gave I to 3%.
disagreement with the values of the observed transition strength
function.
contains
Finally,
if the density of final s t a t e s .already
significant broadening
(Chapter V I ) , the details of
the density of final states will certainly be distorted. These .
effects combine to lessen the significance of the fine s t r u c ­
ture displayed in the density of. final states shown in Fig.
The dashed,
averaged curve for N ^ P t in Fig.
26.■
26 is possibly a
better estimate of N^ pt even though it will not permit as
accurate a recalculation of the optical transitions strength
function as the detailed result will.
C.
Summary
The photoemission data are consistent with the nondirect
4
ODS
(electron/atom -eV)
—•83-"
Fig.
26.
Optical Density of States of
Molybdenum Where the Dashed
Line Is the Average Value for
the Density of States above the
Fermi Energy.
-82transition m o d e l .
The optical and photoemission measurements
of Mo are found to.be in agreement.
located at -0.5 eV,
Peaks in the ODS. of Mo are
-1.6 eV, and -3.9 eV.
tions which produce the large values for
tion, from free electron behavior of
The group of t r a n s i ­
and marked d e v i a ­
in the region 0.5 to
6.0 eV, correlate well with the width of the determined optical
density of states.
The structure’s in
at 1.8 eV and 3.8 eV
correlate well with the peaks at -1.6 eV and -3.9 eV in the
optical density of initial states and with the peaks at
approximately 2.0 eV and 3.8 eV in the optical density of
final states.
reported as
The onset of interband transitions is variously'
.17 eV
(Kirillova,
1967)
and
.44 eV
(Lenham,
1965)
and can be associated with the observed high density of states
at the Fermi energy and possibly with the peak in the density
of initial states at 0.5 eV determined by photoemission studies'.
The quantum yield has a slight decrease at 10.5 eV consistent
with.a relative maximum in the mean absorption depth, onset .
of a volume plasmon,
and a deterioration in the high energy
edge of the EDO's at approximately the same energy.
V.
RUTHENIUM EXPERIMENTAL RESULTS
The results of the optical photoemission, measurements of
ruthenium
(Ru) are summarized in the following c h a p t e r .
results from these measurements
The
are found by arguments very
similar to those given'in detail in the discussion found in
Chapter II.
These same arguments
are given in abbreviated
form the second time.
A.
Optical M e a s u r e m e n t s '
I.
Reflectance.
The reflectance was measured from vapor
deposited films of Ru and are shown in F i g . 27.
Two films
were produced using a sample obtained from the International
Nickel Company
(analysis unknown)
and two films were produced
using a sample from Materials Research Corporation
pure).
The pressure during deposition was
-I x 10
(99.9%
-
8
torr
with a base pressure of =8 x 1 0 _1® torr achieved several
minutes after termination of the deposition.
The r e f l e c t a n c e .
at near normal incidence was extended beyond the LiF window.
cut-off,
11.9 eV, to 14.0 eV by measuring a mechanically
polished bulk sample in a low v a cuum reflectomete.r.
comparison with other measurements
No
of the reflectance or
optical properties of Ru is possible since no published
«*
•optical measurements were found in the literature.
R (hv)
-*$4~
cr'
hv (eV)
(hv) €2(hv)
Reflectance of Ruthenium Vihere
i Is the Angle of Incidence.
hv (gV)
Fig.
22.
Dielectric Function of Ruthenium.
-85Only- one structure point appears
lower energies.
in the reflectance at
A relative minimum- occurs- at 1.4 eV and a
relative maximum at 1.8 eV as seen in Fig.
27.
The reflectance
falls with increasing energy until a second relative minimum
occurs at 11.6 eV.
The optical functions were derived from the reflectance
with an extrapolation of the form R = (1/.E) ^ ^
hv > 14.0 eV.
^ or
The reflectance is monotonically increasing
with photon energy for the highest energies measured.
Since
the assumed form of the high photon energy extrapolation of
the reflectance does not simulate monotonically increasing
b e h a v i o r ' (Appendix A ) , the derived optical functions at the
high energies could be distorted.
2.
Dielectric C o n s t a n t -
The real and the imaginary
parts of the complex dielectric constant are displayed in
Fig.
both
28.
Deviations ,.from free electron-like behavior in
and £2 are noted at between I to 2 eV and 1.3 to 1.5
eV- r e s p e c t i v e l y .
For small photon energies the value of Eg
is large indicating strong absorption.
and Eg is present near 13,0 eV.
A small rise in
This is probably the result
of a poor simulation of the actual reflectance in the region
immediately after 14.0 eV by the assumed extrapolation.
The onset of interband transition, could not be determined
for hv = 0.5 eV.
Comparison of the reflectance of Ru and Mo
-86indicates that the two metals have similar near i n f r a r e d ,
optical properties.
3.
■ Loss F u n c t i o n s .
A surface plasmon is found at 8.7 eV
and a volume plasmon located at 10.2 eV as determined by.the
peak of their respective loss functions as shown in F i g . 29.
The damping due to interband transitions is less in the plasma
region of Ru compared to that of'Mo as t y p i f i e d .by the smaller
values for
or larger amplitude of the loss function in
that same region.
2
4.
a(hv)
and
•and Fig.
A l p h a ; (hv)n(hv),(hv)h(hv)
and
(hv)
(hv)
31 respectively.
2
e2 (hv) .
The optical functions
'
"
e2 (hv) are displayed in Fig.
30
Near 6.0 eV, a(hv) begins to fall
and has a relative minimum at 10.5 eV, indicating the mean
excitation depth of the electrons has a relative maximum at
10;5 eV.
The optical function
(hv)n(hv)
with energy from 2.0 to 10.0 eV.
•(hv)
Both
is nearly a constant
(hv)n(hv)
E2 (hv) begin a very steep rise near 10.0 eV,.
and
These
functions will be used in the interpretation of the p h o t o ­
emission measurements
5. ' Sum R u l e s .
later in this c h a p t e r .
The interband
(Ngp), and volume plasmon
Fig.
32.
(Nv p ) sum rules are presented in
Ru has e.ight electrons
'and d - s h e l l s .
(Njb) , surface plasmon
7 I
(4d 5s ) in the outer s-
The small value observed for N jp at the
highest photon energy indicate that saturation of this sum
-Sy-
0.00
hv (eV)
Loss Function of R uthenium
IxIO -G
hv (eV)
Fig.
30.
Absorption Coefficient c< for
Ruthenium.
—88—
(AS)
(AM) S(A-M)
hv (eV)
Fig.
31.
Optical Functions
for Ruthenium.
Sum Rules (eiectron/aiom)
(hv)^ C^(hv)
(hv)n(hv) and
IO
IR
hv (eV)
Fig.
32.
Interband N rvif Surface Plasma N g p f
and Volume Plasma Nyp Sum Rules
for Ruthenium
-89rule would not occur until a much higher energy.
The
effective- number of electrons participating in the surface
and volume plasihons is small below their measured values of
8.7 eV and 10.2 eV r e s p e c t i v e l y .
For
hv ^ 10 e V , the i n t e r ­
band transitions certainly are the dominant optical process.
The effective number of electrons participating in collective
excitations becomes significant compared to those participating
>
in interband transitions for hv - 10 eV.
This, energy corre- •
lates well with the sharp rise, in the optical function
(hv)n(hv)
B.
and
(hv) ^
(hv).
Photoemission Measurements
Photoemission measurements were made on vapor deposited
films of Ru under high vacuum conditions.
The best evaporation
_g ■
pressure was approximately I x 10
of 8 X 10~
torr with a base pressure
torr achieved within several minutes of the
completion of the deposition.
Ten films were produced from
the two metal samples used in the optical measurements.
.
An
attempt was made to study the electronic structure of 'Ru close
to the Fermi energy by lowering the photoelectric work f u n c ­
tion with a layer of Cs.
The results of these investigations
were not consistent-, with the results obtained from clean Ru
and. are discussed in Appendix B .
All the structures
in the. EDO's
in the present study
-90reproduced well except for a very minor peak which appeared at
E - hv - -2.8 eV in the EDO's from several of the films
produced under poor vacuum conditions
(=10'^ t o r r ) .
The
amplitude, of the small structure in the EDO's was observed
to increase with time.
..Since the peak at E - hv = -.2.8 eV
did not appear in the initial measurements of the samples
produced under better vacuum c o n d i t i o n s , it was assumed to
be.a result of surface contamination.
Only the data taken
under the better vacuum conditions are discussed below.
I.
Quantum Y i e l d .
absorbed photon)
The absolute quantum yield
(electrons/ '
for Ru was measured by the procedures o u t ­
lined in Chapter III and is displayed in F i g . 33 along with
the Fowler
(1931) plot used to d e t e r m i n e .the photoelectric
work function of polycrystalline Ru as 5.4 eV.
No comparison
with other photoelectric work function measurements for the
.metal Ru is possible since none were found in the literature.
The quantum yield for Ru increases smoothly with photon
energy.
The rate of increase slows near 9 eV which is
consistent with the appearance of the final peak seen in the
EDO's
(see b e l o w ) .
Another slight break in rate of increase
produces a very minor structure near 11.0 e W
This roughly
correlates with the relative max i m u m in the mean excitation
depth observed at 10.5 'eV implied by the optical" absorption
coefficient measurement.
Quantum
--91-
S.OsV
Fig.
33
6.O0V
7.OoV
Quantum Yield of Ruthenium
-92.
• 2.
Energy Distribution C u r v e s .
curves for .photoemitted electrons
The energy distribution-
from vapor deposited films
of Ru have been obtained as a function of photon energy by the
experimental procedures and equipment explained in Chapter III
Representative samples of the E D C 's for Ru are shown in F i g .
34.
The EDC's are normalized to the quantum yield and plotted
as a function of
E -hv.
position on E -> plots.,
No structure was found
-The
EDC's
indicate the
with -a fixed
three peaks
.
labeled I, 2, and 3 which have E - hv dependence.
the structure plot for R u , shown in F i g . 35, clearly
indicates the E
E - hv - -0.5 eV
- hv dependence of peaks I and 2
and -1.3 eV respectively.
appears to have some E - hv dependence.
Peak
at
3 at -3.6 eV
Peak 3 is produced ■
by states which are too far below the Fermi energy to be
completely removed from the influence threshold functions even
•at the highest available photon energies.
In addition, the
functional dependence of peak 3 is made more difficult to
determine at high photon energies because of the large
fraction of secondaries.
It will be shown that although part
of this structure is due to secondaries,
the amplitude is too
large to be composed totally of secondaries for the scattering
model assumed' as seen in F i g . 39.
Peak 3 tentatively is ■
identified as a structure in the optical density of initial
states.
- 9 3 -
N(E-hv.hv)
electrons /p h o to n - eV
6 x IO
4x10'
2x10
-Q
-5
—4
E - hr
Fig.
34.
-3
-2
-I
0
(eV)
Normalized Energy Distribution
Curves of Ruthenium versus E - hV
(eV)
Energy of Structure
11.0
12.0
hv (eV)
Fig.
35.
Structure Plot for R u thenium
-95Similar to the result observed from Mo,
there is a
deterioration in the population of high kinetic energyelectrons
11 eV.
in the EDO's between the photon energies
10 eV and
This loss of high energy electrons may be associated
with the observed volume plasmon at 10.2 eV and the relative
maximum in the mean photoexcitation depth at 10.5 eV.
<
When the normalized EDO's for h v - 10 eV are multiplied
by
(hv)n(hv) , the relative magnitude of the .resultant .set of
curves' is essentially the same as that shown in F i g . 34-,
since
(hv)n(hv)
is approximately constant in this spectral
range
(cf. F i g . 31).
set of curves begins
photon energy.
For h v ^ 10 eV, the magnitude of the
a sharp rise in amplitude with increasing
This rise is due to the function
(hv)n(hv)
and occurs in the region where the number of electrons
participating in collective excitations begins.to become
comparable with the number involved in interband excitation
as seen in Fig.
32.
Since the sharp rise in the function
(hv)n(hv) may be associated with collective effects which may
not produce photoemitted electrons,
analysis..
it is ignored in the ODS
The .magnitude of the density of initial states
could be made unique only for a few eV near the Fermi energy
because of this difficulty with the optical' functions.
.. 3.
Optical Density of States for .Ruthenium.
The estimated
scattered electron contribution was obtained by using a zeroth
-96approximation to the O D S .
The zeroth approximation to N ? pt
(E -- hv) was determined as indicated in Chapter III.
The
zeroth approximation to N?p t (E-hv) was then divided into the
EDCs
at various photon energies to obtain the effective
ef f
density of the final states IC
(E). A composite of the
shapes of the various N^^^(E)
in Fig.
36.
In general,
functions obtained is presented
these effective densities of final
states for various photon energies were equivalent even, beyond
the energy 10.0,eV where the optical functions
rapidly increasing with hv.
Since
function for all photon energies,
ef f
(hv)n(hv)
is
(E) is the same
the EDO's can be decomposed
into a product of a function that varies as E - hv with one
that varies as E .
The zeroth approximation to N ^ t (E) was calculated by
inverting the optical transition strength expression, 'Eq.
(2-9).
Only the general envelope of the calculation was
considered significant.
The zeroth approximation to the
optical density of states with an appropriate low energy c u t ­
off and high energy extrapolation
(dashed lines)
is summarized
in F i g . .37.
The fraction of scattered secondary electrons for the
EDO's.of Ru was estimated from E q . (2-35) using the zeroth
approximation "to the O D S .
The results of these Calculations
for the measured EDO's of Ru for photon energies of 8.0 eV
N(E)3'^ (a rb itra ry units)
hv= 8eV
hv = IOsV
hv = i IeV
E (eV)
Fig.
36.
Effective Density of Final States for Ruthenium
arbitrary
unit
—98—
J_-I-.
4 6
-IO -8 -6 -4 -2 0 2
E (eV)
Fig.
37.
8
10
Zeroth Approximation for the
Optical Density of States
Used to Estimate the Scattered
Electron Contribution to the
Energy Distribution Curves of
Ruthenium.
-99and 11.0 eV are displayed in Fig.
38 and Fig.
39 respectively
The scattering correction in Ru is very similar to that found
in Mo except that for a given photon energy the scattering
correction for Ru is less than that assigned to Mo.
The
smaller scattering correction is a result of the higher work
function of Ru.
The .structure which is located at E -. cj> - 2.1 eV in F i g .
39
(peak 3 on Fig.
34)
scattering c o r r e c t i o n .
is attenuated but not removed by the
This peak is therefore at least
partially due to a peak in the initial O D S .
The deviation
of peak 3 from the equal increment rule is possibly explained
by the scattering correction since the peak in the secondary
electron energy distribution falls
in the same kinetic energy
range on the E D O ’s and does obey the equal increment rule
Fig.
38 and. Fig.
39).
(cf
In spite of above evidence indicating
that peak 3 is due to structure in the initial state,
the
reader is cautioned that this evidence depends heavily on
the. low kinetic energy scattered electron correction.
As
noted in Chapter II, the scattering correction for the low
kinetic electron is only approximate,
and the assignment of
peak 3 to a structure in the initial states remains tentative
The final approximation to N ^ P t (E-Iiv) was obtained from
the corrected EDO's by the procedure explained in Chapter TI.
With this estimate of N?p t , the optical transition strength
-100-
electrons/photon-eV
4 x IO
N (E-<£,8)
measured
corrected
scattered
O
l
2
3
4
5
E - ^ (eV)
Fig.
38.
Measured, Corrected, and
Scattered Energy Distribution
of Photoemitted Electrons from
R uthenium at h1r = 8.0 eV.
N (E- <£,11) electron/ photon-eV
-IOl-
4x10
2x10
— measured
— corrected
.• . scattered
E- <£ (eV)
Fig.
39.
Measured, Corrected, and Scattered
Energy Distribution of Photoemitted
Electrons from Ruthenium at hv=11.0eV.
-102integral was inverted again to obtain
are displayed in Fig.
(E).
40 with the appropriate low energy c u t ­
off and consequent high energy extrapolation
line).
These functions
(dotted solid
No further iterations of the ODS analysis technique
were deemed reasonable.
C.
Summary
The photoemission data are consistent with the nondirect
transition model.
The optical constants were obtained from
reflectance studies between 0.5 eV and 14.0 eV and the optical
density of states between ±4.5 eV estimated from photoemission
studies of high vacuum vapor deposited films of Ru.
peaks are seen in the ODS at -0.5 eV,
Definite
-1.3 e V , and 1.5 eV.
A
third peak appears at -3.6 eV, but is not observed over a
sufficient spectral range to uniquely determine its character
as nondirect.
The peaks at -1.8 eV and 1.5 eV correlate well
with structure in
constants
near the same energy.
The optical
indicate a surface and volume plasmon at 8.7 eV and
10.2 eV respectively.
The quantum yield has a very minor
structure near 11.0 eV roughly correlating with the relative
minimum in a'(hv) at 10.5 eV and a sudden degradation of the
population of the high energy electrons in the EDO's.
-COT-
ODS (electrons/atom - eV)
E (eV)
Fig.
40.
Optical Density of States for Ruthenium.
I
INTERPRETATION AND CONCLUSIONS
VI.
The main goal of any experimental investigation is i n t e r ­
preting the experimental data in terms of fundamental physical
processes.
The goal is inevitably accomplished by invoking
some simplifying assumptions
and models.
The simplifications
and assumptions of the present study are mainly documented in
Chapter II with a few appearing in the "results" chapters,
and V.
The predictions
IV
of the simplifying assumptions and
models used to explain the optical and photoemission m e a s u r e ­
ments are mostly summarized in the derived optical density of
states
(ODS).
The ODS of Mo and Ru are discussed and compared
with pertinent theoretical calculations and experimental data.
In addition,
suggestions for future work are included in the
present chapter.
A.
Optical Density of States and Band Structure of Molybdenum
and Ruthenium
I.
Molybdenum.
The ground state density of states as a
function of energy may be determined from energy band c a l c u l a ­
tions.
No energy band calculations outside the region of the
Fermi energy have been published for Mo.
Matthesis
has calculated the b a n d .structure of Mo's
5d counterpart. W,
and Connolly
(1965)
(1968) has calculated Mo's 3d counterpart,
Cr.
Since Cr, Mo, and W have the same crystal structure, body
-105centered. cubic
tions, 3 d ^ 4 s \
(bcc), and similar outer electron shell conf i gura­
4d^5s"*", and •5d^6s ^ r e s p e c t i v e l y , the rigid band
model predicts similar.electronic structure.
•'
The 4d electronic
structure of Mo is more compact than the 5d electronic structure
of W and less compact than the 3d electronic structure of Cr
according to estimates
given by Matthesis
(1965).
calculated the energy separation between the
He has
, and
points in the Brillouin zone for Cr, Mo, and W as a measure of
their d-band widths.
9.2 eV,
He found for Cr, Mo, W^,
and
; 6.9 eV,
10.5 eV, and 14.1 eV respectively, where the results
for Wj and W j j are for two different potentials
effect vary the 6s-5d bands energy separation)
(which in
assumed for the
W calculation.
The energy scale of M a t t h e s i s ' DS for Wj was rescaled
using the above figures and is compared with the experimental
ODS of Mo in. F i g . 41.
Fig.
The dotted portion of the ODS shown in.
41 and on other figures throughout this chapter is less
reliable because of the effects of s e c o n d a r i e s , the threshold
function,
and spectral l i m i t a t i o n s .
There are three peaks
below"the Fermi energy in the experimental ODS
and in the calculated OS
(I, 2, and 3)
(a, b , and c ) . , The energy position
of peaks I and 2 .correspond well with a and c . The most
serious discrepancy noted is in the region near the Fermi
energy
(-1 = E = O
eV).
The calculated DS has a wide
(=1 e V ) ,
- 1 0 5 -
electrons /atom - eV
electrons/atom - eV
— Mo ODS
( left scale)
— Mo DS estimated
from MofthesisDS
for Wj (right scale)-
Fig.
41.
The Optical Density of States of ''olybdenum
Is Shown by the Solid Line.
The Lashed
Line Indicates the Density of States Esti­
mated from IIatthesis Tungsten C1/-,-) Band
Structure Calculations.
-107deep valley and a small value at the Fermi energy
electrons/atom-e V ) .
(Np - 0.6
The small number of states at the Fermi
energy.in the DS is confirmed by specific heat measurements
corrected for electron-phonon interactions
In contrast,
the ODS indicates
(McMillan, 1968).
a peak centered at E = -0.5 eV
and a larger number of states at the Fermi energy
(Np - .9
electron/atom-eV) .,
The ODS and DS of Mo both have one prominent peak above
the Fermi energy.
There are three smaller peaks
in the ODS
for 0 eV < E = 2.5 eV while two are seen near I eV in the
calculated D S .
The structure in the ODS below the Fermi energy is not
as sharp as that displayed in the calculated D S .
The apparent
broadening observed in the ODS of Mo is consistent, with the
broadened structure found in other photoemission investigations
of transition metals
(cf. Ruthenium below; Yu and Spicer,
and Eastman,
This br o a d e n i n g is principally due to the
1969).
finite lifetime of the photoexcited state
1964),
(Berglund and Spicer,
and possibly due to many body interactions
and Phillips,
1967;
(Spicer,
1967
1965).
The density of states obtained from energy band c a l c u l a ­
tions of M o 1s 3d and 5d counterpart,
Cr and W respectively,
indicate the rigid band model is applicable to the VI B .g r o u p .
Both Cr and W have three similarly shaped peaks
in the DS
-108below the Fermi energy
(cf. Fig.
42 and Fig.
41 r e s p e c t i v e l y ) .
The smallest and broadest peak appears at the lowest energy
(peak a ) , the largest peak appears at an intermediate energy
(peak b ) , and the intermediate amplitude peak appears nearest
the Fermi energy
(peak c ) .
As seen in Table II, the energy
position of these calculated peaks
in.the density of states
of Cr and W are simple related by a nearly constant scale
factor close to that predicted by Matthesis
(.66).
TABLE II
Peak
Cr (Connolly,
1968)
W t (Ma t t h e s i s ,
1 1965)
Scale
Factor
a
-1.3 eV
-1.8 eV
.72
b
-2.4 eV
-3.2 eV
.75
C
-3.3 eV
-4.5 eV
.73
The experimentally obtained photoemission data of the
group VI B 3d and 4d transition metals, Cr and Mo respectively,
are not simply related by the rigid band model.
three structure points
curves in Fig.
positions
in the ODS of Cr and Mo
42 and Fig.
There are
(cf.
the solid
41, respectively) but their energy
are not related by a simple constant, scale factor
as displayed in Table IIIv
- 1 0 9 -
O
I" OPfiCAU pa
I?)
P5
-5
-4
-3
- z.
-1
ENERGY (=V)
9ig,
42.
The Optical Density of States
from Eastman (i960) ( Solid L i n e )
and the Density of States Estimated
from Connolly *(1968) Band Structure
Calculations for Chromium (Dashed
L i n e ).
-110TABLE III
Peak
Mo (present Cr
study)
(Lapeyre and
Kress , 1968)
C r (Eastman,
1968)
Average
Scale
Factor
I
-0.5 eV
-0.2 eV
-0.4 eV
.60
2
- 1.6 eV
-1.1 eV
- 1.2 eV
.72
3
-3.9 eV
-2.2 eV
-2.3 eV
.58
F u r t h e r m o r e , inspection of F i g . 43 reveals that the
amplitude of the ODS of Mo cannot be derived from that of Cr.
This is particularly true of the region -2 eV ^ E < 0 eY where
the ODS of Mo and Cr should be well determined by photoemission
studies.
The amplitude differences between the experimentally
observed O D S 1s and the calculated OS's of Mo and Cr may
indicate the ODS is simply not a good replica of the u n p e r ­
turbed ground state density of states.
The optical transi­
tion probability matrix elements for the transition metals
may have E - hv dependence since their wave functions are
localized o r .antibonding near the top of the d-bands and are
diffuse or bonding near the bottom of the d-bands
et-al,
1967).
(Cuthill,
Using ,the dipole approximation Cuthill et al
calculated large matrix element variations due to these
systematic changes
in the calculated wave functions of bcc Fe
-Ill-
— Mo
— Cr
E (eV)
Fig.
43.
Comparison of the Optical
lonsity of States of Molybdenum
(present s t u d y ) and Chromium
(3s atman, 1968 ),
-112from the top to the bottom of the d-ban-d.
Although their
simple estimate of large matrix element variations is not
supported by experimental studies
(Eastman,
1968) , their
calculation does demonstrate the possibility of E - hv
dependence in the matrix element of the optical transition
probability as discussed in Chapter
II e E - hv dependence in
the matrix, elements cannot be distinguished from the ground
state density of initial states effects
optical studies and,
therefore,
in photoemission and
cannot be detected in the
present study.
An interesting comparison between the estimated DS
obtained from W and the ODS of Mo is obtained by ad hocadjustment of the d-band width and the position of the Eermi
energy.
These parameters of the DS were adjusted for "best"
fit with the O D S .
The energy scale of M a t t h e s i s '
whs
reduced by a factor of 0.92 and the Fermi energy moved
approximately one eV.
The good agreements
The result is displayed in Fig.
44.
in the energy positions and amplitudes of
all the structures below the Fermi energy may be fortuitous.
W i t h o u t ■detailed knowledge of the effects of the various
parameters used in the energy band calculations
(eg. s-d
energy bands s e p a r a t i o n ) ; it is difficult to estimate the
plausibility and significance of the above comparison.
- 1 1 3 -
trons /a to m - eV
--- Mo OD S
( le f t scale)
------- Mo DS estimated
from Matthesls DS
for VVn
p
( right
K
>
CD
I
irons / a t
E
CD
CD
E (cV)
Fig.
44.
The Optical Density of States of
Molybdenum Compared with the Density
of States Estimated from Matthesis’
Tungsten (7TT) Band Structure Calculation,
Note the Energy Scale and the Position
of the Fermi Level of Matthesis* ^ t t
Calculation Was Arbitrarily Adjusted,
-1142.
R u t h e n i u m .■ It is generally felt that the DS c a l c u ­
lated from band structure results is sensitive to the crystal
structure of the metal
(Matthesis, 1969).
the ODS of hexagonal close packed
with the DS of hep Re
part
(5d^6s^)
7 1
(bcc) F e (3d 5s ) in Fig.
(hep) Ru
For this reason
7 I
(4d 5s ) is compared
instead of with its 3d counter 45.
assuming the rigid band m o d e l .
'
The comparison was made
The scale factor of .75 was
used to adjust the energy axis of the Re D S .
The energy scale
factor was estimated from the energy eigenvalue differences
between various symmetry points
in the Brillouin zone for Re
compared to the same points for Ru.
The scale factor was
chosen with the aid of some unpublished energy eigenvalue,
calculations of Ru kindly supplied by Matthesis
The results presented in Fig.
(1969).
45 indicate that the •
observed optical density of states, below -2\0 eV is broadened
and averaged compared with the estimated density of states.
This is similar to the results obtained in Mo and is probably
due to the same effects suggested there.
A major discrepancy
exists between the ODS and the calculated DS near the Fermi
energy
( - 2 . 0 -eV = E = 0.0 e V ) .
The calculated DS shows a
large valley in the density of states approximately 2 eV wide.
■This discrepancy is quite similar to that found in the c o m p a r i ­
son of the ODS of Mo with that expected from the DS of.W.
discussion given there would presumably apply h e r e .
The
(electrons/atom-
XO
O
O
ZO
SlT-
E
o
I
Fig.
45.
The Optical Density of States (Solid Line) and
The Density of States of Ruthenium Estimated
from Matthesis’ Rhenium Band Structure Calcu­
lation (Dashed Line).
-116Excellent agreement between the ODS and the calculated
estimate beyond the Fermi energy is found.
empty states indicates
The shape of the
the termination of the d-band near
4.5 eV and the beginning of a new series of states in the
same region.
These could possibly be the beginning of the.
5p s t a t e s .
The hypothesis that the density of states is more s e n s i ­
tive to crystal structure than to the atomic structure of the
atom in the free state is tested in F i g . 46 where the- ODS of
Ru is compared with the density of states of its 3d atomic
structure counterpart non-magnetic Fe calculated from Wood's
energy band results by Matthesis
(1965).
■ The energy scale of
the Fe density of states was expanded by the same factor
Matthesis
gives for the relative d-band width of Cr to Md
since Fe and Ru are similarly situated on the periodic table.
There
is no detailed agreement between the optical density of
states and the calculated density of states obtained from n o n ­
magnetic Fe above or below the Fermi energy.
Although the
similarities between the ODS for Ru and the calculated density
of states for Re are in reasonable agreement o n l y , they are
certainly in better agreement than the ODS and the calculated
density of states for non-magrietic iron as the original
hypothesis
i n d i c a t e d ..
The structure observed in the photoemission EDO's and
- 1 1 7 -
Ru ODS
(le ft scole)
( e le c f r o n s / o to m - sV)
(electrons / a t o m - eV )
Ru DS estimated
from W oods DS
for Fe
(right sccle)
E (eV)
i'lg.
46 .
The Optical Density of States of
Ruthenium Compared with the Density of
States Estimated f r o m M a t t h e s i s 1
Calculation of the Density of States
of nonmagnetic iron from food's Energy
Band Calculations.
-118optical constants of hep Ru are not as well defined or as
numerous as those observed in bcc Mo.
Since the hep structure
is not as spherically symmetric as cubic structures,
the
poorly defined structures in the optical and photoemission
data of hep Ru could be a result of the assumption that
po l y c r y s t a l line samples of hep metals may be treated as if they
had isotropic properties.
observed differences
This possible explanation for the
in the optical and photoemission p r o p e r ­
ties of the hep and bcc transition metals could be tested by
studies of simple crystal s a m p l e s .
"3.
Nondirect and Direct Transitions in Molybdenum and
Ruthenium.
The photoemission data from Mo and Ru were found
to be consistent with the nondirect transition model with the
possible exception of the sudden attenuation in the high
kinetic energy peaks of the EDO's.for hv > 10 eV
and Fig.
34).
(cf. F i g . 19
The disappearance of this high energy peak
could indicate direct transition character to this peak in
the EDO's of both Mo and Ru.
The lack of agreement between '
the measured ODS and estimated DS just below the Fermi energy
may also indicate direct transitions
in the data.
In addition to the implications of the above questionable
predictions of the nondirect transition model applied to theexperimental data for the metals Mo and Ru, recent detailed
calculations' for the metal Cu indicate direct and nondirect
--119transitions cannot bs unambiguously distinguished in the photo
emission data because of some degeneracy in the predictions
the direct and nondirect transition model
1969).
of
(Smith and Spicer,
The photoemission data from the 3d bands of Cu were
calculated from the energy bands assuming direct transitions
and constant matrix elements.
The calculations of Smith and
Spicer were successful in predicting the existance and energy
position of the four principle pieces of structure observed in
the d-bands of the experimental E D G 1s of Cu.
The calculation
did not successfully predict the amplitude of the structures
in the E D C s
of Cu.
This latter failure of the calculation,
according to. Smith and Spicer, may have been a result..of the
assumption of constant matrix elements. ■
The above discussion .indicates additional investigations
.are needed into the character of the optical transitions of Mo
and Ru.
One obvious
investigation would be the extension of
the energy band calculations
so that the photoemission E D C s
could be calculated for Mo and Ru.
Such calculations would be
particularly interesting since the occupied energy bands of
Mo and Ru, as indicated by the energy band of W and Re
respectively,
are quite different in shape compared to the
relatively flat
of Cu.
(slowly varying with Bloch wave number) bands
-120B.
The Optical Density of States and Isotropic Mass Effect in
Molybdenum and Ruthenium
The superconducting transition temperature,
mass dependent
for many' of the simple metals.
Tc , is isotopic
The isotopic
mass effect is a direct consequence of assuming that the
attractive interaction.needed for the theory of superconduc­
tivity is a result of an electron-phonon interaction and is
summarized by
M a T f, = constant
(6-1)
where M is the atomic mass of the metal and the exponent a
is 1/2 for simple metals which have a normal isotopic mass
dependence.
This relation is not verified for some of the
transition metals .(Garland, 1963).
a
is found to be
1961),
and
.00 ± .15 for Ru
.37 ± .04 for Mo
In particular,
the constant
(Geballe and M a t t h e s i s ,
(Matthesis et al,
1963)..
These
anomalous results lead some investigators to suggest that
interactions other than the virtual exchange of phonons should
be considered in the theory of superconductivity
et al,
1961) . . Others,
(e.g. Geballe
reluctant to give up the interaction
that worked well for. the simple metals, have suggested models
for the s- and d-band electronic densities of states of the
■
transition metals which would explain the deviations from the .
normal isotopic mass effect.
Garland
(1963) was able to show
from investigation of the Coulomb s-s and s-d electron
-121interactions that such effects were possible in any transition
metal having, a narrow
(<< I eV) peak in the -density of states
near the Fermi energy and a small s -band Fermi wave number.
Nordtvedt
(1965)
investigated the short wave length longitu­
dinal phonon frequencies and found similarly that an anomalous
isotopic effect was possible
if the effective mass of electrons
in the d-bands. is m^. > ■10 ,.m e 2.e c tr on an^
is large near
(-0.2 eV)
density, of d- states
the Fermi energy.
According to the models proposed by Garland and N o r d t v e d t ,
a large narrow peak in the density of electronic states should
appear near the Fermi energy of Ru and Mo.
The experimental
O D S 1s obtained in- this study do not indicate peaks at the
Fermi energy.
The peaks nearest to the Fermi energy are -
-0.4 eV and -0.5 eV for Ru and Mo respectively.
Furthermore,
the observed peaks near the Fermi energy are not as narrow
as those suggested by the models of Nordtvedt and Garland;
In .addition to the above experimental evidence for
anomalous electronic structure
in Ru and Mo, McMillan
(1968)
has shown that a more refined treatment than the BCS theory
assumed by Garland and N o r d t v e d t , will account for the
quenched isotopic mass effect without any anomalous conditions
being imposed on. the density of states.
His results for both
Ru and Mo indicate a small density of electronic states at the
Fermi e n e r g y .
McMillan concludes
that the electron-phonon
-122coupling c o n s t a n t s , which are directly related to the isotopic
•mass effect,
depend mainly on the phonon frequencies and are
insensitive to a large variation in the electronic properties,
i.e.
the band-structure density of states.
His results are
probably"the best statement of the effects implied by the
anomalous
isotopic mass effect in Ru and Mo since he used a
later "state of the art" theory.
Contrary to the above experimental and theoretical
implications,
there is some evidence in the
constants data published by Kirillova et al
infrared optical
(1967)
of a sharp
narrow structure in the density of states near E p .
Using their
results for n(hv) and k(hv), the optical transition strength
O
function (hv)
e2 (hv) = 2n(hv)k(hv) was calculated and is
presented, in Fig.
47.
Their data implies the existence of an
intense narrow band of transitions very near the Fermi energy.
The transitions cannot be unambiguously associated with the
initial or final states from the optical data alone
(Chapter
II), but they do fall in the narrow energy range predicted by
Garland and Nordtvedt and are probably beyond the resolution
of photoemission studies.
The above ambiguities indicate more experimental studies
are needed.
Since a narrow peak in the DS may be beyond the
resolution of the present photoemission study,
the infrared
optical functions of Mo should be reinvestigated to confirm
- 1 2 3 -
Optical
Transition Function
(hy)';'2(hv)
(eV)
(Kirillova Gt a I)
Fig.
47.
Optical Transition Strength Function
Calculated from Infrared Optical Constants
of Molybdenum Measured.by Kirillova et al.
-124the strange behavior of the optical constants measured by
Kirillova et.al.
would be helpful
In a d d i t i o n ,.the optical constants of Ru
in determining if the narrow band of t r a n s i ­
tions seen in Mo is a general property of the transition
metals with anomalous
isotopic mass
effects.
APPENDIX
APPENDIX A
Kramers-Kronig Analysis
The complex amplitude of an electromagnetic wave reflected
from a metal surface is found from Maxwell's equations to be
=
=
where R(hv)
(N-H)
(N+l)
(A-I)
(A-2)
IR (hv) I e,"1 ^
is the reflectance- at normal incidence,
cj) is the
phase change of the reflected e l e c t r o m a g n e t i c 'field and
N(hv)
= n(hv)
+ ik(hv),
(Born and Wolff , 1964) . - Since f
is a complex analytic function of the photon energy there
exists a dispersion relation between the real and the
imaginary parts which may be w ritten as
(Matthews and.Walker,
1964)
In R(E)
$ (hv)
'o
With <f)(hv) known,
dE
(A-3)
.E^ - (hv)^
the optical constants n(hv)
and k(hv) -may
be calculated since separating the real and imaginary part
of Eq.- (A-I)
n(hv)
■ I - R(hv)____________
1 + R(hv)
2 R(hv) cos <j)(hv)
(A-4)
k(hv)
_______ -2 R(hv)
I + R (hv) ^ ”2
(Ar%)
sin $(hv)___
R (hv)- cos <j)(hvj.
-127All the optical constants may be calculated from this set
(Chapter II).
To calculate the function cf>(hv) extrapolations of the
reflectance must be assumed since the reflectance cannot be
measured over the entire range of the definite integral.
The
reflectance at the lower photon energies may be extrapolated
to zero energy by the Hagen-Rubeiis relation
(Z i m a n , 1964).
R(hv) - I - /v/'a
where- a Q is the dc conductivity.
(A-6)
The calculation for <j)(hv)
is not sensitive, to the low energy extrapolation region since
In R (hv)
O as hv -> O .
The Hagen-Rubens relation was used to
extrapolate the reflectance to zero energy considering oQ as
a parameter adjusted to fit R (hv) at 0.5 e V . '
The high energy region extrapolation has a more profound
effect on the values calculated for ■(J)(hv) since In R (hv) does
not become small as R (hv) decreases.
Studies made by others '
indicate that the final results for cj)(hv) are n o t very sensitive
to the functional form assumed for the extrapolation if only
functional forms with one adjustable parameter are considered
(Veshe, 1967).
A guide to reasonable high photon energy
extrapolation may be found by considering the results of the
high energy reflectance from a free electron metal,
fV
Rfree(h v )
16
i.e.
'i
JSl
(A-7)
-128
where
is the plasma f r e q u e n c y .
Considering the above result
the high energy extrapolation functional form considered in
this work was chosen to be
(A-8)
R(hv)
where R(hvn ) is the last measured value for the reflectance
at the photon energy hvn .
This choice produces
a smooth" fit
in magnitude but not necessarily a smooth fit to the shape or
first derivative
(cf. Chapter V).
The contribution to <p(hv) due to the high energy ex t r a ­
polation may be determined exactly.
into Eq.
Substituting Eq.
(A-8)
(A-3) and solving
X
-CO
hv
hv
cf)(hv)
'
n In R(E)
'o
dE
E 2 - (hv)2
^
dE
in R(E)
, hv
Jhv
E 2 - (hv)2
(A-9)
hv
-
£
TT
*A - -»“
I
£
m = l 'm
w
hv
hv
hv
hv
n
+ hv
A - 10)
hv
hv
The high energy extrapolation reduces to the determination of
.the single parameter x.
A.computer program was written to choose the value of x
which best fit the r e f l e c t a n c e ■at some angle other than normal
-129incidence.
(A-I) .
A flow diagram of the program is given in Fig.
.Initially x was assigned the value 4.0'0‘ and (|>(hv)
calculated.
With this <j)(hv) the reflectance at. angle ^ could
be calculated from the generalized Fresnel reflection equations
(Hunter, 1965).
^
Rc (Qihv)
(a-cos6-) ^
(a+cos0) ^ +■ b^
X
I ■+ (a-sin9 t a n ^6)^ + b^
(A-Il).
(a+s in 6 t a n ^ 6)^ + b^-
[ (n2 - k 2 - Sin 2G )2 "+ 4 n 2k 2 ]^
7
7
7
(A-12)
+ (nz' - k z - sin 0)
[ (n2 - k 2 - S i n 2G )2 + 4n2k 2 ]^
(n
7
7
7
(A-13)
- k z - sin 9)
The calculated reflectance R c (9 ,hv) was. then compared to the
measured Rm (0,hv)
computed,
and the rms deviation between the two sets
x was then adjusted to reduce the rms deviation.
The cycle was continued-until the adjusted x agreed with the
previous x to three significant figures.
criterion,
When x satisfied this
the functions' n (hv) and k (hv) .were calculated and
all the optical functions d e t e rmined and plotted.
• .
-13 O-
Reflectance data
at normal incidence
Reflectance data
at angle 0
Calculate (f> t hen n
and k normal
incidence data with
x = 4*0
Calculate R (e hv)
for given x
. Find rms deviation
fro m m e a sured Rm (8 ,hv)
Correct x to reduce
the rms deviation
Does x agree with
previous x to three
Calculate and graph
Fig.
A-I.
Logic D i a g r a m of Computer Program
Used to Calculate Optical Constants
-131The computer program was checked by calculating the
optical functions' of Cu from the published reflectance values
(Ehrenreich and Philipp,
1962).
Good agreement was. found
between the optical functions calculated assuming x ■= 4.0,
and the published optical functions'.
APPENDIX B
Study of Cesium Covered Ruthenium
The surface barrier and consequently the photoelectric
work function of a metal is reduced when covered with a
monolayer of the active metal cesium
(Cs) (Decker, 1954).
The reduced work function allows photoemission investigation,
of an extended energy range provided the Cs layer does not
disturb the intrinsic photoemission properties of the metal
(e.g.
Berglund and Spicer,
1964.and Callcott and M a c R a e , 1969)
The photoemission properties of Cs covered films of Ru were
investigated in the present study using in addition to the
basic apparatus and techniques previously described,
a Cs
source.
A Cs gun was obtained which produced Cs + ions under high
vacuum conditions
(Weber, 1966) .
The Cs
ions were attracted
to the Ru coated photocathode by a small negative
potential.
(- -10 v)
The total Cs coverage was monitored by measuring .
the ion current to the photocathode.
only a small current
(~ 10
The gun would deliver
a m p ) , consequently the time
required to produce a monolayer Was on the order of hours.
The technique produced a reasonably uniform coverage of Cs on
the photocathode since there was
little variation of the photo
current when the photocathode was probed with a light beam of
a few m m ^ cross section.
-133The quantum yield per absorbed photon of Cs covered Ru
is presented in Fig.
(B-I).
The reflectance of clean Ru was
used to correct the experimentally observed yield per incident
photon.
The coverage of Cs on the photocathode for the data
shown was approximately two monolayers.
sharply from the low photon energies
The yield rises
and slows
its rate of
increase producing a shoulder between 4 eV and 5 eV.
The
quantum yield rises rapidly again with increasing photon
energy and levels off after approximately 8 eV.
Within the
range of measurements, hv = 11.9 eV, few additional excitations
which produce photoemitted electrons
The low photon energy E D C s
appear above 8 eV.
did not display the structure
or character of the initial states seen in the photoemission
results of clean Ru.
Since the. photoemission properties of -
clean Ru were not seen,
the Cs layer may have alloyed with the
•
Ru and altered the density of states or the Cs layer could
have strongly scattered the photoemitted electrons as they
passed through the surface.
The high photon energy E D C s
in Fig.
of Cs covered Ru are presented
(B-2).
The E D C s are plotted as a function of E - hv
i
'
so that structure due to the initial states should be
stationary
(Chapter II).
and another near -8.4 eV.
are extremely attenuated.
One large peak appears near -4-. 2 'eV
The high energy portions of the E D C s
Ru(Cs)
Quantum Yield
(electrons/absorbed photon)
-134-
Fig. B-I
Quantum Yield of Ruthenium with Approximately 2
Monolayer of Cesium.
Ru ( Cs)
Arbitrarily
Normalized
Cl-
(-Ml 'AAj-3) N
/
-
8.8
10.0
E - h-v (eV)
Fig. B-2.
LDCs of Photoemitted Electrons from Cs covered Ru
-136The observed photoemission properties may be explained if
a surface or volume plasmon which produces discrete energy
losses of
- 4.2 eV is postulated.
The first peak at E - hv - 4.2 eV is a result of electrons
losing a discrete energy E ^ .
The second peak at E - hv
is caused.by electrons which experience two losses,
8.4eV
2E^, to the
plasmon or is possible due to simple scattering of the high
kinetic energy electrons.
The latter effect is more likely the
cause of the -8.4 eV peak in the EEC's since the yield saturates
near 8.0 eV indicating few new transitions beyond this energy.
The extreme attenuation of the high energy portions of the EEC
indicates the probability of exciting this plasmon loss is
rather high,
i.e. most of the electrons which escape interact
inelastically with the plasmon.
The energy of the losses observed indicates they cannot
be associated with the volume plasmon of Ru,
2.8 e V .(Mayer and H i e t e l , 1966)
these two metals
(hv^^^ce
=
10.2 eV or Cs,
or the surface plasmons of.
, Arakawa,. 1966):
The losses are probably due to the interaction of the Ru and
the Cs layer in the region of or on the .surface of the p h o t o ­
cathode.
The energy loss may occur during the transport of the
photoemitted electrons.
Alternately the loss could occur
,
during the optical absorption process as has been proposed :for
I
-137'Cs covered transition metals
(Callcott and M a c R a e , 1969):
The data presented here cannot distinguish between these
possibilities.
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MONTANA STATE UNIVERSITY LIBRARIES
3 I
IIIIIIIII
10010 95£
D378
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cop.2
Photoemission and
optical investigation of
the electronic structure.
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